Confined subgroups in groups with contracting elements

Inhyeok Choi June E Huh Center for Mathematical Challenges
Korea Institute for Advanced Study
Seoul, South Korea [email protected] , Ilya Gekhtman , Wenyuan Yang Beijing International Center for Mathematical Research
Peking University
Beijing 100871, China P.R. [email protected] and Tianyi Zheng Department of Mathematics, University of California San Diego, 9500 Gilman Dr. La Jolla, CA 92093, USA. [email protected]

(Date: May 4th 2024)

Abstract.

In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to conformal measures. In particular, we prove that confined subgroups are conservative, and examples of subgroups with nontrivial Hopf decomposition are constructed. We show a connection between Hopf decomposition and quotient growth and provide a dichotomy on quotient growth of Schierer graphs for subgroups in hyperbolic groups.

Key words and phrases:

confined subgroups, Patterson-Sullivan measures, contracting elements, Hopf decomposition, growth rate

2000 Mathematics Subject Classification:

Primary 20F65, 20F67, 37D40

(I.C.) Supported by Samsung Science & Technology Foundation (SSTF-BA1301-51), Mid-Career Researcher Program (RS-2023-00278510) through the NRF funded by the government of Korea, KIAS Individual Grant (SG091901) via the June E Huh Center for Mathematical Challenges at KIAS, and by the Fields Institute for Research in Mathematical Sciences.

(W.Y.) Partially supported by National Key R & D Program of China (SQ2020YFA070059) and National Natural Science Foundation of China (No. 12131009 and No.12326601)

1. Introduction

Let $G$ be a locally compact, second countable topological group. The space $\mathrm{Sub}(G)$ of all closed subgroups in $G$ , equipped with the Chabauty topology, is a compact metrizable space on which $G$ acts continuously by conjugation. Measurable and topological dynamics of the action $G\curvearrowright\mathrm{Sub}(G)$ have been instrumental in recent advances in the study of lattices in semi-simple Lie groups, see [1, 29]. In particular, on the measurable dynamics side, invariant random subgroups (IRS), which are invariant measures on $\mathrm{Sub}(G)$ , have attracted a lot of interest since the terminology was coined in [2]. Their topological counterpart, namely $G$ –minimal systems in $\mathrm{Sub}(G)$ , were introduced as uniformly recurrent subgroups (URS) in [34]. URS play an important role in the connections between reduced $C^{\ast}$ –algebras and topological dynamics developed in [43, 15].

The notion of confined subgroups was introduced in [37] in the study of the ideal structure of group rings of simple locally finite groups. More recently, it has been further investigated in [13, 14] for a variety of countable groups of dynamical origin. Let $\Gamma,H$ be two subgroups of $G$ , where $H$ is not necessarily contained in $\Gamma$ . We say that $H$ is confined by $\Gamma$ in $G$ if its $\Gamma$ –orbit (i.e. all conjugates of $H$ under $\Gamma$ ) does not accumulate in $\mathrm{Sub}(G)$ to the trivial subgroup. Equivalently, there exists a compact confining subset $P\subseteq G$ such that $g^{-1}Hg\cap P\setminus\{1\}\neq\emptyset$ for all $g\in\Gamma$ . When $\Gamma=G$ , we simply say that $H$ is a confined subgroup of $G$ . Every subgroup $H$ in a non-trivial URS of $\Gamma$ is confined by $\Gamma$ . Conversely, if $H$ is confined by $\Gamma$ , then the $\Gamma$ –orbit closure of $H$ in $\mathrm{Sub}(G)$ contains a non-trivial $\Gamma$ –minimal system.

In a semi-simple Lie group, discrete confined torsion-free subgroups can be described by the geometric condition that the action on the associated symmetric space has bounded injectivity radius from above. Fraczyk-Gelander [29] showed that in a higher-rank simple Lie group, discrete confined subgroups are exactly lattices, confirming a conjecture of Margulis. This is obviously false in the rank-1 case: indeed, normal subgroups of uniform lattices are confined in both the lattice and the ambient Lie group. Instead, Gelander posed a conjecture on the growth rates of discrete confined subgroups. This conjecture was recently proved by Gekhtman-Levit [31]. The methods of [29, 31] have probabilistic ingredients, using stationary random subgroups to derive geometric results.

The main goal of the present article is to investigate confined subgroups in a general class of discrete groups $\Gamma$ , which acts on a geodesic metric space $(\mathrm{X},d)$ with contracting elements defined as follows.

Definition 1.1.

¹¹1There are several notions of contracting elements in the literature. Our notion here is usually called strongly contracting by other authors. As there are no other ones used in this paper, we’d keep use of this terms to be consistent with [71, 72]

An isometry $g\in\mathrm{Isom}(\mathrm{X})$ is called contracting if the orbital map

n\in\mathbb{Z}\longmapsto g^{n}o\in\mathrm{X}

is a quasi-isometric embedding, so that the image $A:=\{g^{n}o:n\in\mathbb{Z}\}$ has contracting property: any ball $B$ disjoint with $A$ has $C$ -bounded projection to $A$ for a constant $C\geq 0$ independent of $B$ .

Contracting elements are usually thought of as hyperbolic directions, exhibiting negatively curved feature of the ambient space. The main examples we keep in mind with application include the following:

Examples 1.2.

•

Fundamental groups of rank-1 Riemannian manifolds with a finite geodesic flow invariant measure of maximal entropy (the BMS measure), where contracting elements are exactly loxodromic elements.
•

Hyperbolic groups; more generally, relative hyperbolic groups, where loxodromic elements are contracting elements.
•

CAT(0)-groups with rank-1 elements, including right-angled Coxeter/Artin groups, where contracting elements coincide with rank-1 elements.
•

Mapping class groups on Teichmüller space endowed with Teichmüller metric, where contracting elements are exactly pseudo-Anosov elements.

On a heuristic level, one might ask to what extent confined subgroups are large, or resemble normal subgroups. To be more precise, we consider two closely related aspects: the dynamics of a discrete subgroup $H<\mathrm{Isom}(\mathrm{X},d)$ acting on a suitable boundary of the space $\mathrm{X}$ ; and the growth rates of $H$ and its quotient $\mathrm{X}/H$ . Our investigation focuses mainly on the questions related to these aspects for a discrete confined subgroup $H$ .

Fix a basepoint $o\in\mathrm{X}$ . We define the growth rate (or critical exponent) of the orbit $Ho$ as

(1)

\omega_{H}=\limsup_{n\to\infty}\frac{\log\sharp N_{H}(o,n)}{n},

where $N_{H}(o,n)=\{go:g\in H,d(o,go)\leq n\}$ . When $\Gamma$ is discrete, we also consider the growth rate $\omega_{\Gamma/H}$ , defined below in (3) analogously to (1), of the image $\Gamma o$ under the projection $\mathrm{X}\to\mathrm{X}/H$ equipped with the quotient metric. ²²2In the literature, if $H$ is a subgroup of $\Gamma$ , $\omega_{H}$ is usually called relative growth; the relative growth of a normal subgroup $H$ is referred as cogrowth relative to the growth of $\Gamma/H$ (particularly on $\Gamma=\mathbf{F}_{d}$ , see [33]). Occasionally, the terminology is reversed: $\omega_{\Gamma/H}$ is called cogrowth relative to the growth $\omega_{\Gamma}$ of $\Gamma$ (see [53, 8]). We find that the former is more convenient in this paper.

These growth rates have been intertwined with other quantities in the study of the geometric and dynamic aspects of discrete groups for a long history. In rank-1 symmetric spaces, the famous Elstrodt-Patterson-Sullivan-Corlette formula relates growth rate to the bottom $\lambda_{0}$ of Laplace-Beltrami spectrum on $\mathrm{X}/H$ as follows:

\displaystyle\lambda_{0}(\mathrm{X}/H)=\begin{cases}\omega_{H}(h(\mathrm{X})-\omega_{H}),&\omega_{H}\geq h(\mathrm{X})/2\\ \frac{h^{2}(\mathrm{X})}{4},&\omega_{H}\leq h(\mathrm{X})/2\end{cases}

where $h(\mathrm{X})$ is the Hausdorff dimension of visual boundary. If $\Gamma$ is a lattice, then $h(\mathrm{X})=\omega_{\Gamma}$ .

In the discrete setting, by the cogrowth formula due to Grigorchuk, for $H<\mathbf{F}_{d}$ a subgroup in the free group on $d$ generators, the spectral radius $\rho(\mathrm{X}/H)$ of simple random walks on $\mathrm{X}/H$ is given by

\displaystyle\rho(\mathrm{X}/H)=\begin{cases}\frac{\sqrt{2d-1}}{2d}(\frac{\sqrt{2d-1}}{\mathrm{e}^{\omega_{H}}}+\frac{\mathrm{e}^{\omega_{H}}}{\sqrt{2d-1}}),&\mathrm{e}^{\omega_{H}}\geq\sqrt{2d-1}\\ \frac{\sqrt{2d-1}}{d},&\mathrm{e}^{\omega_{H}}\leq\sqrt{2d-1}\end{cases}

where $\mathrm{X}$ is the $2d$ –regular tree (the standard Cayley graph of $\mathbf{F}_{d}$ ), and $\log(2d-1)$ is the Hausdorff dimension of the space of ends of $\mathrm{X}$ .

The seminal works of Kesten [44] and Brooks [16] characterize amenability in terms of spectral radius of random walks and Laplace spectrum on Riemannian manifolds, respectively. Using the above formulae, these results can be interpreted in a common geometric setting: for $\Gamma$ of some natural classes of groups, a normal subgroup $H<\Gamma$ attains the maximal relative growth rate $\omega_{H}=\omega_{\Gamma}$ if and only if the quotient $\Gamma/H$ is amenable. This perspective is fruitful, with the latest results in [20, 21] for strongly positively recurrent actions (SPR) or statistically convex-cocompact actions (SCC) on hyperbolic spaces. In a different direction, Kesten’s theorem is generalized beyond normal subgroups: for IRS in [3] and URS in [28]. We refer the reader to these papers and reference therein for a more comprehensive introduction. To put into perspective, we draw in Fig. 1 the relation between various actions including SPR and SCC actions considered in this paper.

The question of whether the inequality $\omega_{\Gamma/H}<\omega_{\Gamma}$ holds for normal subgroups $H\neq\Gamma$ has been actively investigated [6, 62, 26] since the introduction of the notion of growth tightness in [33]. The most general results currently available for normal subgroups of divergence type actions are given by [5, 71], while the situation for general subgroups $H$ remains largely unexplored in the literature. Furthermore, the relations between $\omega_{H},\omega_{\Gamma}$ and $\omega_{\Gamma/H}$ still remain mysterious, despite recent works [39, 19].

In this paper, for a confined subgroup $H$ of $\Gamma$ , we establish the conservativity of the boundary action of $H$ , a strict lower bound $\omega_{H}>\omega_{\Gamma}/2$ (such an inequality is sometimes referred to as cogrowth tightness), and an inequality relating $\omega_{H}$ to $\omega_{\Gamma}$ and $\omega_{\Gamma/H}$ . Our approach studies boundary actions equipped with conformal measures and relies on geometric arguments with contracting elements. Notably, we do not rely on any input from random walks or considerations of probability measures on the Chabauty spaces. We first present some applications before stating our general results.

1.1. Main applications

Before presenting them in full generality, we state some of our main results in several well-studied geometric settings.

First of all, let us consider a proper geodesic Gromov hyperbolic space or a CAT $(0)$ space $\mathrm{X}$ , equipped with the Gromov or visual boundary $\partial{\mathrm{X}}$ in the first and second cases respectively. Let $\Gamma<\mathrm{Isom}(\mathrm{X})$ be a non-elementary discrete subgroup containing a loxodromic or rank-1 element accordingly. If the action $\Gamma\curvearrowright\mathrm{X}$ has purely exponential growth (or more generally, of divergence type), there exists a unique $\omega_{\Gamma}$ –dimensional Patterson-Sullivan measure class $\mu_{\mathrm{PS}}$ on the Gromov or visual boundary $\partial{\mathrm{X}}$ constructed from the action $\Gamma\curvearrowright\mathrm{X}$ . Denote by $E(\Gamma)$ the set of isometries in $\mathrm{Isom}(\mathrm{X})$ which fix pointwise the limit set $\Lambda(\Gamma o)$ (Definition 5.5).

Theorem 1.3.

Assume the action $\Gamma\curvearrowright\partial{\mathrm{X}}$ has purely exponential growth. Let $H<\mathrm{Isom}(\mathrm{X})$ be a nontrivial torsion-free discrete subgroup confined by $\Gamma$ with a compact confining subset. Then $\omega_{H}\geq\omega_{\Gamma}/2$ . Furthermore,

(1)

If $H$ preserves the measure class of $\mu_{\mathrm{PS}}$ , then the action $H\curvearrowright(\partial X,\mu_{\mathrm{PS}})$ is conservative.
(2)

If $H$ admits a finite confining subset intersecting trivially $E(\Gamma)$ , then $\omega_{H}>\omega_{\Gamma}/2$ and $\omega_{H}+\omega_{\Gamma/H}/2\geq\omega_{\Gamma}$ .

In many circumstances, the group of isometries fixing boundary pointwise $\partial{\mathrm{X}}$ is trivial (or finite), e.g. if $\mathrm{X}$ is a CAT(-1) space or a geodesically complete CAT(0) space with a rank-1 geodesic (i.e., bounding no half flat). In case of $\Lambda(\Gamma o)=\partial{\mathrm{X}}$ , $E(\Gamma)$ is finite and the assumption in the item (2) is always fulfilled.

An important subclass of (uniquely) geodesically complete CAT $(0)$ spaces are provided by Hadamard manifolds (i.e. a complete, simply connected $n$ –dimensional Riemannian manifold with non-positive sectional curvature). Its visual boundary $\partial\mathrm{X}$ , which is defined as the set of equivalence classes of geodesic rays, is homeomorphic to $\mathbb{S}^{n-1}$ . Let $\Gamma<\mathrm{Isom}(\mathrm{X})$ be a torsion-free discrete group. The quotient manifold $M=\mathrm{X}/\Gamma$ is called rank-1 if it admits a closed geodesic without a perpendicular parallel Jacobi field. In other words, $M$ is rank-1 if and only if $\Gamma$ contains a contracting element [12, Theorem 5.4].

Theorem 1.4.

Let $M=\mathrm{X}/\Gamma$ be a rank-1 manifold, and assume that the geodesic flow on the unit tangent bundle $T^{1}M$ has finite measure of maximal entropy. Let $\mu_{\mathrm{PS}}$ be the unique $\omega_{\Gamma}$ –dimensional Patterson-Sullivan measure class on $\partial\mathrm{X}$ constructed from the action $\Gamma\curvearrowright\mathrm{X}$ . Let $H<\mathrm{Isom}(\mathrm{X})$ be a nontrivial torsion-free discrete subgroup confined by $\Gamma$ with a compact confining subset. Then $\omega_{H}\geq\omega_{\Gamma}/2$ . Furthermore,

(1)

If $H$ preserves the measure class of $\mu_{\mathrm{PS}}$ , then the action $H\curvearrowright(\partial\mathrm{X},\mu_{\mathrm{PS}})$ is conservative.
(2)

If $H$ admits a finite confining subset intersecting trivially $E(\Gamma)$ , then $\omega_{H}>\omega_{\Gamma}/2$ and $\omega_{H}+\omega_{\Gamma/H}/2\geq\omega_{\Gamma}$ .

As noted above, if $\Gamma\curvearrowright\mathrm{X}$ has full limit set $\Lambda(\Gamma o)=\partial{\mathrm{X}}$ , then $E(\Gamma)$ is trivial.

A few remarks are in order on the statements and their background. We refer the reader to the subsequent subsections for more detailed discussions.

Remark 1.5.

•

The fundamental group $\Gamma$ is not necessarily a lattice in $\mathrm{Isom}(\mathrm{X})$ . The maximal entropy measure lies in the measure class $\mu_{\mathrm{PS}}\times\mu_{\mathrm{PS}}\times\mathbf{Leb}$ modulo $\Gamma$ –action on $\partial\mathrm{X}\times\partial\mathrm{X}\times\mathbb{R}$ (usually called the Bowen-Margulis-Sullivan measure in the pinched negatively curved case [25] or Knieper measure in the rank-1 case [45]). The finiteness of this measure can be characterized in several ways (e.g. [25] and [61]).

If $\Gamma$ is a uniform lattice and $\mathrm{X}$ is a rank-1 symmetric space of noncompact type, then $\mu_{\mathrm{PS}}$ is the Lebesgue measure on $\partial\mathrm{X}$ invariant under $\mathrm{Isom}(\mathrm{X})$ (see [58]). In this restricted setting, if $H$ is confined by $\mathrm{Isom}(\mathrm{X})$ (equivalently by any uniform lattice, see Lemma 5.3), the strict inequality $\omega_{H}>\omega_{\Gamma}/2$ was proved by Gekhtman-Levit [31] (without the finite confining subset assumption).
•

The item (1) is known in confined Kleinian groups for the Lebesgue measure on the sphere ([52, Theorem 2.11]). However, its proof does not generalize to Gromov hyperbolic spaces. Our proof is different and works in general metric spaces; see Theorem 1.11.
•

If $H$ is contained in $\Gamma$ , the assumptions after “Furthermore” are redundant, and the corresponding conclusions hold without them. A confined subgroup in a non-uniform lattice $\Gamma$ is not necessarily confined in $\mathrm{Isom}(\mathrm{X})$ , so the inequality in item (2) does not follow from [31].

Another application is given for confined subgroups in the mapping class group $\mathrm{Mod}(\Sigma_{g})$ of a closed orientable surface $\Sigma_{g}$ ( $g\geq 2$ ). The finitely generated group $\mathrm{Mod}(\Sigma_{g})$ is actually the orientation preserving isometry group of the Teichmüller space $\mathcal{T}_{g}$ with Teichmüller metric, on which it acts properly with growth rate $(6g-6)$ . Investigating the similarities between $\mathrm{Mod}(\Sigma_{g})$ and lattices in semi-simple Lie groups has been an active research theme. The following result fits into the rank-1 phenomenon of $\mathrm{Mod}(\Sigma_{g})$ .

Theorem 1.6.

Consider the measure class preserving action of $G=\mathrm{Mod}(\Sigma_{g})$ on the Thurston boundary $\mathscr{PMF}$ of the Teichmüller space $\mathcal{T}_{g}$ with Thurston measure $\mu_{\mathrm{Th}}$ (cf. [7] recalled in Example 2.26).

Let $H<G$ be a nontrivial confined subgroup. If $g=2$ , assume $H$ is infinite. Then the following holds

(1)

$H\curvearrowright(\mathscr{PMF},\mu_{\mathrm{Th}})$ is conservative.
(2)

$\omega_{H}>\omega_{G}/2=3g-3$ .
(3)

$\omega_{H}+{\omega_{G/H}}/{2}\geq 6g-6$ .

The items (2) and (3) for normal subgroups were proved by Arzhantseva-Cashen [4], and Coulon [19] respectively. The item (1) is new even for the normal subgroup case.

Remark 1.7.

The Mumford compactness theorem implies that $G$ acts cocompactly on the $\epsilon$ –thick part of $\mathcal{T}_{g}$ for any fixed $\epsilon>0$ . It follows that in the setting of Theorem 1.6, $H$ is a confined subgroup of $G$ if and only if $\mathcal{T}_{g}/H$ has bounded injectivity radius from above over any fixed thick part of $\mathcal{T}_{g}$ .

1.2. Ergodic properties of boundary actions

It is classical that any action of a countable group equipped with a quasi-invariant measure admits the Hopf decomposition into the conservative and dissipative parts. In 1939, Hopf found the dichotomy in the geodesic flow of Riemannian surface with constant curvature that the flow invariant measure is either ergodic (thus conservative) or completely dissipative. Hopf’s result was extended to higher dimensional hyperbolic manifolds by Sullivan [64], forming a still-growing collection of the now-called Hopf-Tsuji-Sullivan dichotomy results in a number of settings by many authors. For the convenience of the reader, we state the HTS dichotomy for conformal measures on the Gromov boundary for a discrete group on CAT(-1) space ([61]):

Theorem 1.8 (HTS dichotomy for CAT(-1) spaces).

Assume that $H\curvearrowright\mathrm{X}$ is a proper action on CAT(-1) space. Let $\{\mu_{x}:x\in\mathrm{X}\}$ be a $\omega$ -dimensional $H$ -conformal density on the Gromov boundary $\partial{\mathrm{X}}$ for some $\omega>0$ . Then we have the dichotomy: either the following equivalent statements hold:

(I.1)

$\mathcal{P}_{H}(s,x,y)=\sum_{h\in H}\mathrm{e}^{-sd(x,hy)}$ is divergent at $s=\omega$ .
(I.2)

$\mu_{x}$ is supported on the set of conical points $\Lambda^{\mathrm{con}}(Ho)$ .
(I.3)

$\mu_{x}\times\mu_{x}$ is ergodic. In particular, $\mu_{x}$ is ergodic.

or the following equivalent statements hold:

(II.1)

$\mathcal{P}_{H}(s,x,y)=\sum_{h\in H}\mathrm{e}^{-sd(x,hy)}$ is convergent at $s=\omega$ .
(II.2)

$\mu_{x}$ is null on the set of conical points $\Lambda^{\mathrm{con}}(Ho)$ .
(II.3)

$\mu_{x}\times\mu_{x}$ is completely dissipative.

Under the set (I) of conditions, we must have $\omega=\omega_{H}$ and $\mu_{x}$ is unique up to scaling.

This dichotomy and its variants have then been established for conformal densities on the visual boundary of CAT(0) spaces [46]; and in our context, groups with contracting elements acting on the convergence boundary $\partial{\mathrm{X}}$ in a sense in [72]. See Coulon’s recent works [19, 22] for a more complete statement on horofunction boundary.

The notion of convergence boundary $\partial{\mathrm{X}}$ (Definition 2.13) provides a unified framework for the following boundaries equipped with a conformal density (cf. Definition 2.24 and Examples 2.15):

Examples 1.9.

•

The prototype example is the rank- $1$ symmetric space $\mathrm{X}$ of non-compact type, with a family of $\mathrm{Isom}(\mathrm{X})$ –equivariant conformal measures $\{\mu_{x}:x\in\mathrm{X}\}$ in the Lebesgue measure class on the visual boundary of $\mathrm{X}$ . This class of conformal measures can be realized as Patterson-Sullivan measures for any uniform lattice $\Gamma<\mathrm{Isom}(\mathrm{X})$ . See [58].
•

The Gromov boundary of a hyperbolic group, with quasi-conformal density constructed first by Patterson [54] from a proper action of $G$ on a hyperbolic space $\mathrm{X}$ , and then studied extensively by Sullivan [64, 66, 65] in late 1970s, with a long array of results [18, 61] by many authors, to name a few.
•

Visual boundary of a CAT(0) space admitting a proper action with rank-1 elements. The Patterson’s construction equally applies to produce a conformal density on visual boundary.
•

Thurston boundary of Teichmüller space with a conformal density constructed from Thurston measures (in the Lebesgue measure class) on the space of measured foliations in [7]. The conformal density could be realized as Patterson-Sullivan measures. See Example 2.26 for more details.
•

In recent works [19, 72], the horofunction boundary $\partial_{h}{\mathrm{X}}$ has been proven to be a convergence boundary for any proper action of a group with a contracting element; furthermore, on the horofunction boundary, a good theory of conformal density can be developed: in particular, the Shadow Lemma and HTS dichotomy are available.

Suppose that the space $\mathrm{X}$ admits a convergence boundary $\partial{\mathrm{X}}$ (or keep one of these examples in mind). Motivated by rank-1 symmetric space, we assume that there is an auxiliary proper action $\Gamma\curvearrowright\mathrm{X}$ , in order to endow a $\omega_{\Gamma}$ –dimensional $\Gamma$ –equivariant conformal density $\{\mu_{x}:x\in\mathrm{X}\}$ on $\partial{\mathrm{X}}$ . Sometimes, we assume that $\Gamma\curvearrowright\mathrm{X}$ is statistically convex-cocompact in the sense of [71] (SCC action; see Definition 2.38); this is the case in Theorem 4.2. Among the groups in Example 1.2, the first three classes are known to admit SCC actions; and all of these examples have purely exponential growth (PEG action):

(2)

\forall n\geq 0:\;\sharp N_{\Gamma}(o,n)\asymp\mathrm{e}^{n\omega_{\Gamma}}.\quad

Most of results actually only assume a PEG action $\Gamma\curvearrowright\mathrm{X}$ , or even a proper action of divergence type (DIV action) in a greater generality. The class of DIV actions is naturally featured, as one of the two alternatives, in the HTS dichotomy 1.8. To be clear, we shall make precise the actions $\Gamma\curvearrowright\mathrm{X}$ assumed in our results stated in what follows, and refer to Fig. 1 for the implication between these various actions.

Our first main object is to analyze the Hopf decomposition for a measure-class-preserving action of a group $H$ on $(\partial{\mathrm{X}},\mu_{o})$ .

In contrast to the HTS dichotomy of geodesic flow invariant measures, which can be applicable to the $H$ –action on $(\partial{\mathrm{X}}\times\partial{\mathrm{X}},\mu_{o}\times\mu_{o})$ , the ergodic behavior of actions on one copy of the boundary, which is intimately linked with the horocylic flow [42], exhibits more complicated situations. For Kleinian groups, Sullivan [65, Theorem IV] characterizes the conservative component with respect to the Lebesgue measure on the boundary in terms of the (small) horospheric limit set. This leads to several important applications in Ahlfors-Bers quasi-conformal deformation theory and Mostow-type rigidity theorems ([65, Sect. V &VI]). In a recent work [35], Grigorchuk-Kaimanovich-Nagnibeda carried out a detailed analysis of the boundary action of a subgroup $H$ in a free group $\mathbf{F}_{d}$ with respect to the uniform measure on $\partial\mathbf{F}_{d}$ . These works serve as a source of inspiration in our considerations in a more general framework.

The limit set $\Lambda(Ho)$ denotes the set of accumulation points of $Ho$ in a convergence boundary $\partial{\mathrm{X}}$ . In the last two cases in Example 1.9, $\Lambda(Ho)$ may depend on $o\in\mathrm{X}$ . To avoid this difficulty, a nontrivial partition $[\cdot]$ on $\partial{\mathrm{X}}$ was introduced so that the set $[\Lambda(Ho)]$ of $[\cdot]$ –classes on $\Lambda(Ho)$ is independent of the choice of base points. Analogously to the notions in the actions of the convergence group, we can define conical points and the big/small horospheric limit points. More details and precise definitions are provided in Section 2.

Figure 1. Relations between assumptions on actions under consideration. The equivalence in the second column is expected to hold in groups with contracting elements (some known cases as cited).

First, we present a criterion when the action of a subgroup is completely dissipative. This is a vast generalization of [35, Theorem 4.2] for free groups, which can be traced back to the case of Fuchsian groups [55] (see [49]).

Theorem 1.10.

Let a group $H<\mathrm{Isom}(\mathrm{X})$ act properly with contracting elements, and $\mu_{0}$ be the PS measure constructed from a proper SCC action $\Gamma\curvearrowright\mathrm{X}$ . Assume that $\omega_{H}<\omega_{\Gamma}/2$ . Then $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=0$ , where $\Lambda^{\mathrm{Hor}}{(Ho)}$ denotes the big horospheric limit set of $Ho$ defined in (2.46).

In addition, assume that $H$ is torsion-free and preserves the measure class of $\mu_{o}$ on $\partial{\mathrm{X}}$ (in particular, if $H$ is a subgroup of $\Gamma$ ). Then the action of $H$ on $(\partial{\mathrm{X}},\mu_{o})$ is completely dissipative.

Note that, if $\mathrm{X}$ is hyperbolic and $\Gamma\curvearrowright\mathrm{X}$ is co-compact, then any $H<\mathrm{Isom}(\mathrm{X})$ preserves the measure class of $\mu_{o}$ (see Lemma 4.1). We expect Theorem 1.10 to hold under the weaker assumption that $\Gamma\curvearrowright\mathrm{X}$ has purely exponential growth (see Remark 2.45 on particular situations where this is true).

Next, we prove that the action of a confined subgroup is conservative. Denote by $E(\Gamma)$ the set of isometries in $\mathrm{Isom}(\mathrm{X})$ which fix every $[\cdot]$ –class in the limit set $[\Lambda(\Gamma o)]$ (Definition 5.5).

Theorem 1.11.

Suppose that a discrete subgroup $H<\mathrm{Isom}(\mathrm{X})$ is confined by $\Gamma$ with a compact confining subset $P$ . Assume that $\Gamma\curvearrowright\mathrm{X}$ is of divergence type, and furthermore:

(i)

Either $H$ is torsion-free and $\mathrm{X}$ is a Gromov hyperbolic or geodesically complete CAT(0) space, or
(ii)

the confining subset $P$ is finite, and intersects trivially $E(\Gamma)$ .

Then $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=1$ . In particular, $H\curvearrowright(\partial{\mathrm{X}},\mu_{o})$ is conservative.

Note that the action of $H$ on $(\partial{\mathrm{X}},\mu_{o})$ is not necessarily ergodic, with examples found in free groups [35] and Kleinian groups [52]. As an immediate corollary, we obtain the following non-strict inequality. With a further inequality in Theorem 1.15, it turns out that the assumption of the SCC action can be relaxed to be the DIV action.

Corollary 1.12.

Assume that $\Gamma\curvearrowright\mathrm{X}$ is SCC. In the setting of Theorem 1.11, we have $\omega_{H}\geq\omega_{\Gamma}/2$ .

1.3. Growth inequalities for confined subgroups

Via a different approach, we can upgrade the result in Corollary 1.12 to a strict inequality. We work with PEG actions $\Gamma\curvearrowright\mathrm{X}$ strictly larger than the SCC action considered in §1.2 (particularly in Theorem 4.2).

Theorem 1.13.

Assume that the proper action $\Gamma\curvearrowright\mathrm{X}$ has purely exponential growth with a contracting element. If $H<\mathrm{Isom}(\mathrm{X})$ is a discrete subgroup confined by $\Gamma$ with a finite confining subset that intersects trivially with $E(\Gamma)$ , then $\omega_{H}>\omega_{\Gamma}/2$ .

If $H<\Gamma$ is normal, Coulon [19] and independently the third named author [72] proved this inequality using boundary measures in greater generality (only assuming $\Gamma\curvearrowright\mathrm{X}$ is of divergence type). In addition, if $H$ is a normal subgroup of divergence type, then $\omega_{H}=\omega_{\Gamma}$ . Whether this holds for general confined subgroups remains open.

Together with Corollary 1.12, Theorem 1.13 has the following application to a finite tower of confined subgroups of $\Gamma$ .

Corollary 1.14.

Assume that $\Gamma\curvearrowright\mathrm{X}$ is a PEG action with contracting elements. Let $H:=H_{0}<H_{1}\cdots<H_{s}=:\Gamma$ be a sequence of subgroups so that $H_{i}$ is confined in $H_{i+1}$ for each $0\leq i<s$ . Then $\omega_{H}>\omega_{\Gamma}/2^{s}$ .

This in particular applies to an $s$ –subnormal subgroup $H$ for some integer $s\geq 1$ , for which there exists a strictly increasing subnormal series, $H_{0}:=H\unlhd H_{1}\cdots\unlhd H_{s}=:\Gamma$ , of length $s$ . This statement was known in free groups by Olshanskii [53], who also proved that the lower bound is sharp: for any $\epsilon>0$ and $s\geq 1$ , there exists some $s$ –subnormal subgroup for which $\omega_{H}<\epsilon+\omega_{\Gamma}/2^{s}$ .

Next, consider the space $\mathrm{X}/H:=\{Hx:x\in\mathrm{X}\}$ equipped with the quotient metric. That is, given $Hx,Hy\in\mathrm{X}/H$ , define $\bar{d}(Hx,Hy)=\inf\{d(x,hy):h\in H\}$ . Denote by $\pi:\mathrm{X}\to\mathrm{X}/H$ the natural projection.

Fixing a point $o\in\mathrm{X}$ , consider the ball-like set in the image of $\Gamma o$ in $\mathrm{X}/H$ :

N_{\Gamma/H}(o,n)=\{Hgo:\bar{d}(Ho,Hgo)\leq n\}

and define its growth rate $\omega_{\Gamma/H}$ as follows:

(3)

\omega_{\Gamma/H}=\limsup_{n\to\infty}\frac{\log\sharp N_{\Gamma/H}(o,n)}{n}.

As $\Gamma o$ and $\Gamma o^{\prime}$ have a finite Hausdorff distance, the growth rate $\omega_{\Gamma/H}$ does not depend on $o\in\mathrm{X}$ . If $\mathrm{X}$ is the Cayley graph of $\Gamma$ and $H$ is a subgroup of $\Gamma$ , then $\mathrm{X}/H$ is the Schreier graph associated with $H$ . If in addition, $H$ is a normal subgroup of $\Gamma$ , then $\omega_{\Gamma/H}$ is the usual growth rate of $\Gamma/H$ with respect to the projected generating set, which explains the notation.

We obtain the following inequality relating the growth and co-growth of confined subgroups.

Theorem 1.15.

Assume that the proper action $\Gamma\curvearrowright\mathrm{X}$ is of divergence type with a contracting element. If a discrete group $H<\mathrm{Isom}(\mathrm{X})$ is confined by $\Gamma$ with a finite confining subset that intersects trivially with $E(\Gamma)$ , then

\omega_{H}+\frac{\omega_{\Gamma/H}}{2}\geq\omega_{\Gamma}

The inequality for normal subgroups was obtained earlier by Jaerisch and Matsuzaki in [39] for free groups, and by Coulon [19] in a proper action of divergence type with contracting elements. Our proof uses a shadow principle for confined subgroups, from which we deduce the inequality following closely Coulon’s proof. Compared to their works, it is worth noting that the confined subgroup $H$ is not required to be contained in $\Gamma$ .

As a corollary, we can relax the SCC action in Corollary 1.12 to be DIV actions.

Corollary 1.16.

In the setup of Theorem 1.15, we have $\omega_{H}\geq\omega_{\Gamma}/2$ .

Note that it follows from Theorem 1.15 that if $\Gamma/H$ has sub-exponential growth (i.e. $\omega_{\Gamma/H}=0$ ), then $\omega_{H}=\omega_{\Gamma}$ . By [19][Cor. 4.27], for a normal subgroup $H$ , we have $\omega_{H}=\omega_{\Gamma}$ if $\Gamma/H$ is amenable. We do not know what condition on $\Gamma/H$ might characterize the equality $\omega_{H}=\omega_{\Gamma}$ for more general confined subgroups.

We are now in a position to give the the proofs of Main Applications in §1.1.

Proofs of Theorems 1.3, 1.4 and 1.6.

Theorems 1.3 and 1.4 follow immediately from Theorems 1.11, 1.13 and 1.15 together. We note here that assumption in (2) are redundant: the elliptic radical $E(\Gamma)<\mathrm{Isom}(\mathrm{X})$ (i.e. fixing $\Lambda(\Gamma o)$ pointwise) intersects $\Gamma$ in a finite subgroup. If $H<\Gamma$ is a torsion-free subgroup, then the confining subset of $H$ intersects trivially $E(\Gamma)$ , so the assumption for (2) holds automatically.

For mapping class groups, if $g>2$ , then $E(G)$ is trivial, so Theorem 1.6 follows exactly as above. If $g=2$ , $E(G)$ contains hyperelliptic involution, then an additional argument is needed to conclude the proof. Indeed, since $G$ is residually finite, we may pass to finite index torsion-free subgroups $\hat{G}$ of $G$ , where $\hat{G}\cap H$ is infinite and torsion-free, so Theorem 1.6 follows. The growth rate remains the same after taking finite index subgroups, so the proof in the general case follows. ∎

1.4. Maximal quotient growth

We now relate the ergodic properties of the boundary actions studied in §1.2 to the growth of a $\Gamma$ orbit $\Gamma o$ in the quotient $\mathrm{X}/H$ . A proper action $\Gamma\curvearrowright\mathrm{X}$ is called growth tight if $\omega_{\Gamma/H}<\omega_{\Gamma}$ holds for any infinite normal subgroup $H<\Gamma$ . In the setting of Theorem 1.11, the action of a torsion-free $H$ on the boundary is conservative. The inequality $\omega_{\Gamma/H}<\omega_{\Gamma}$ in particular implies that the quotient growth of $H\curvearrowright\mathrm{X}$ is slower than $\Gamma\curvearrowright\mathrm{X}$ in the following sense:

\frac{\sharp N_{\Gamma/H}(o,n)}{\sharp N_{\Gamma}(o,n)}\to 0.

Following Bahturin-Olshanski [8], we say that $H\curvearrowright\mathrm{X}$ has maximal quotient growth ⁴⁴4In terms of [8], $N_{\Gamma/H}(o,n)$ is the growth function of the right (usually non-isometric) action of $\Gamma$ on $H\backslash\Gamma$ . The right action is not relevant here, so we take the term of quotient growth instead. if the following holds:

\sharp N_{\Gamma/H}(o,n)\asymp\sharp N_{\Gamma}(o,n).

We shall establish an equivalence of conservative action and slower quotient growth, and a characterization of maximal right coset growth in terms of boundary actions. These are only obtained here for co-compact actions on hyperbolic spaces. We expect the characterizations to hold in a much more general setup. See Question 1.18 and Remark 10.12 for conjectures for an extension to mapping class groups.

Theorem 1.17.

Assume that $\Gamma$ acts properly and co-compactly on a proper hyperbolic space $\mathrm{X}$ and $\mu_{o}$ is the unique Patterson-Sullivan measure class on the Gromov boundary $\partial{\mathrm{X}}$ . Let $H<\Gamma$ be a subgroup. Then the small/big horospheric limit set is $\mu_{o}$ –full, if and only if $\mathrm{X}/H$ grows slower than $\mathrm{X}$ . Moreover, $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})<1$ is equivalent to the following

\sharp\{Hgo:\bar{d}(Ho,Hgo)\leq n,g\in\Gamma\}\asymp\mathrm{e}^{\omega_{\Gamma}n}.

Before giving some corollaries, let us point out the following question which plays a key role in the proof of Theorem 1.17:

Question 1.18.

Let $\mu_{o}$ be a conformal measure on a boundary $\partial{\mathrm{X}}$ without atoms (cf. Example 2.49). Under which conditions, do we have $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})$ ?

This question has been raised in Sullivan’s work [65], where he showed for the real hyperbolic space $\mathbf{H}^{n}$ that a discrete subgroup $H<\mathrm{Isom}(\mathbf{H}^{n})$ is conservative with respect to $(\mathbf{S}^{n-1},\mathbf{Leb})$ if and only if the volume of $\mathbf{H}^{n}/H$ grows slower than $\mathbf{H}^{n}$ . The key part of the proof is that the big and small horospheric limit sets differ in a $\mathbf{Leb}$ –null set. We elaborate this proof to answer the question positively for any subgroup $H$ in Theorem 1.17. See Theorem 10.9 for details.

For normal subgroups, adapting an argument of [48, 27], the question can be answered positively in the following specific situation.

Theorem 1.19.

Assume that $\Gamma\curvearrowright\mathrm{X}$ has purely exponential growth with a contracting element. Let $H$ be an infinite normal subgroup of $\Gamma$ . Then $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})=\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=1$ .

As another application of Theorem 1.17, we obtain a characterization of conservative actions.

Corollary 1.20.

Under the assumption of Theorem 1.17, assume that $H$ is torsion-free. Then the action $H\curvearrowright(\partial{\mathrm{X}},\mu_{o})$ is conservative if and only if the quotient growth of $H\curvearrowright\mathrm{X}$ is slower than $\Gamma\curvearrowright\mathrm{X}$ .

Another corollary is characterizing the purely exponential growth of right cosets as in the “moreover” statement.

Let $H,K<\Gamma$ be two subgroups with proper limit sets $[\Lambda(Ho)],[\Lambda(Ko)]\subsetneq[\Lambda(\Gamma o)]$ (we say these are of second kind following the terminology in Kleinian groups). In [36], it is shown that the double cosets $HgK$ over $g\in\Gamma$ have purely exponential growth:

\sharp\{HgKo:d(o,HgKo)\leq n,g\in\Gamma\}\asymp\mathrm{e}^{\omega_{\Gamma}n}.

If $K=\{1\}$ , this amounts to counting the right coset $Hg$ as above. In hyperbolic groups, this case is thus covered in Theorem 1.17, which furthermore gives a characterization of purely exponential growth of $\{Hg:g\in\Gamma\}$ by $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})<1$ . If $H=\{1\}$ , a characterization remains unknown to us for purely exponential growth of left cosets $gK$ .

To conclude our study, we present some examples of subgroups with nontrivial Hopf decomposition. First such examples are constructed in free groups ([35]). Here we are able to construct these subgroups for any SCC actions on hyperbolic spaces.

Theorem 1.21.

Suppose that $\Gamma$ admits a proper SCC action on a proper hyperbolic space $\mathrm{X}$ . Let $f\in\Gamma$ be a loxodromic element. Then there exist a large integer $n$ so that the normal closure $\llangle f^{n}\rrangle$ contains subgroups of second kind with nontrivial conservative part and dissipative part.

We expect this result extends to any SCC action. Indeed, denoting $G=\llangle f^{n}\rrangle$ , if $\omega_{G}<\omega_{\Gamma}$ is proved, $\mathrm{X}$ could be replaced with a general metric space. Amenability criterion in [21] for hyperbolic spaces says that $\omega_{G}=\omega_{\Gamma}$ is equivalent to the amenability of $\Gamma/G$ . If $n$ is sufficiently large, $\Gamma/G$ is non-amenable, so $\omega_{G}<\omega_{\Gamma}$ . If the amenability criterion holds for any SCC action, then $\omega_{G}<\omega_{\Gamma}$ follows and so does Theorem 1.21.

Figure 2. A flow chart of main results, where the assumptions DIV, PEG and SCC on the action

\Gamma\curvearrowright\mathrm{X}

are illustrated in Fig. 1

1.5. Ingredients in the proofs and organization of the paper

We highlight some tools and ingredients in obtaining the results presented above. The reader may restrict their attention to the special case $H<\Gamma$ for simplicity. Our standing assumption is that the space $\mathrm{X}$ admits a convergence boundary $\partial{\mathrm{X}}$ , and $\Gamma$ acts properly on $\mathrm{X}$ . Readers who are not familiar with the background on convergence boundary might keep in mind the example where $\mathrm{X}$ is hyperbolic space with Gromov boundary $\partial{\mathrm{X}}$ : for this special case we have illustrations of the main ideas throughout the text.

We refer to Fig. 2 for an overview of the main theorems and their logical relations. Fig. 1 illustrates the implications between assumptions on the actions.

Results on Hopf decomposition

The proof of results on the Hopf decomposition of $H\curvearrowright(\partial{\mathrm{X}},\mu_{o})$ , as presented above in §1.2, makes extensive use of the conformal measure theory $\{\mu_{x}:x\in\mathrm{X}\}$ developed in [72] on the convergence boundary, provided that $H\curvearrowright\mathrm{X}$ is of divergence type. The case of horofunction boundary was independently obtained by Coulon [19]. The key facts we use include the following:

(1)

the Shadow Lemma 2.31 holds for $\mu_{x}$ , and
(2)

$\mu_{x}$ is supported on conical points (Lemma 2.35), and is unique up to bounded multiplicative constants on the reduced horofunction boundary (Lemma 2.37).

A well-prepared reader would notice that these are standard consequences of the HTS dichotomy (e.g. for CAT(-1) spaces stated in Theorem 1.8). In greater generality, this has been established for groups with contracting elements as mentioned above in [19, 72]. We recommend the reader to keep in mind the situation of CAT(-1) spaces in the sequel without losing the essentials.

The proof of Theorem 1.10 in the case that $\mathrm{X}$ is a Gromov hyperbolic space is obtained by recasting the combinatorial proof in free groups [35] in appropriate geometric terms. The general case mimics the same outline, with the additional input of the recent result [57] on full measures supported on regularly contracting limit sets. If $H$ is a confined subgroup, we push a generic set of limit points into the horospheric limit set of $H$ to prove the conservativity of the action of $H$ , Theorem 1.11. This strategy has appeared in the works [48, 27] for normal subgroups of groups acting on hyperbolic spaces.

Key technical results on confined subgroups

Toward the proof of Theorem 1.11, we prove the following key technical result stated in Lemma 5.8, which can be viewed as an enhanced version of the Extension Lemma 2.10.

Lemma 1.22 (=Lemma 5.8).

There exists a finite subset $F$ of contracting elements in $\Gamma$ with the following property: for any $g\in\Gamma$ , there exist $f\in F$ and $p\in P$ such that $gfpf^{-1}g^{-1}$ lies in $H$ and

|d(o,gfpf^{-1}g^{-1}o)-2d(o,go)|\leq D

where $D$ depends only on $F$ and $P$ .

To facilitate the reader with perspective from hyperbolic spaces, we include a short section §3 to demonstrate the main ideas and tools without invoking much preliminary material. Complete proofs of Theorem 1.10 and Theorem 1.11 are provided for this special case.

This lemma provides the geometric property of confined subgroups that allows us to prove statements in a manner similar to that of normal subgroups. In particular, adapting the argument in [72] for normal subgroups, we prove the following.

Lemma 1.23 (Shadow Principle in §7).

Let $\{\mu_{x}\}_{x\in\mathrm{X}}$ be a $\omega_{H}$ –dimensional $H$ –quasi-equivariant quasi-conformal density supported on $\partial{\mathrm{X}}$ . Then there exists $r_{0}>0$ such that

\begin{array}[]{rl}\|\mu_{y}\|e^{-\omega_{H}\cdot d(x,y)}\quad\prec_{\lambda}\quad\mu_{x}(\Pi_{x}(y,r))\quad\prec_{\lambda,\epsilon,r}\quad\|\mu_{y}\|e^{-\omega_{H}\cdot d(x,y)}\\ \end{array}

for any $x,y\in\Gamma o$ and $r\geq r_{0}$ .

The usual Shadow Lemma asserts only the above inequalities on a smaller region of $x,y\in Ho\subseteq\Gamma o$ , where $\|\mu_{y}\|=1$ . Hence, the Shadow Principle provides useful information in the case that $\Gamma o$ is a much larger set. If $\Gamma\curvearrowright\mathrm{X}$ is cocompact, the above result holds over the whole space $\mathrm{X}$ .

Cogrowth tightness of confined subgroups

The nonstrict part of inequality $\omega_{H}>\omega_{\Gamma}/2$ asserted by Theorem 1.13 follows immediately from Theorems 1.10 and 1.11. The strict part turns out to require a more subtle argument. The strategy follows the outline of the proof of strict inequality for normal subgroups of divergence type in [72](§8). The proof for the convergence type is easier, so we omit the discussion here.

The main point in [72] is to obtain a uniform bound $M$ on the mass of $\mu_{y}$ over $y\in\Gamma o$ :

(4)

\forall go\in\Gamma o,\quad\|\mu_{go}\|\leq M

Once this is proved, the coefficient $\|\mu_{y}\|$ in the Shadow Principle disappears, so we would obtain $\omega_{H}\geq\omega_{\Gamma}$ from a standard covering argument. That is, the number of shadows $\Pi_{o}(go,r))$ , $go\in A_{\Gamma}(o,n,\Delta)$ which contain a given point of $\partial X$ is bounded by a uniform constant $C$ , and therefore

e^{-\omega_{H}n}\sharp A_{\Gamma}(o,n,\Delta)\leq C

which yields $\omega_{H}\geq\omega_{\Gamma}$ and thus $\omega_{H}=\omega_{\Gamma}$ follows for normal subgroups $H$ of divergence type.

To make life easier, let us assume $\mathrm{X}$ is a CAT(-1) space, so the set (I) of conditions in Theorem 1.8 holds. As $H\curvearrowright\mathrm{X}$ is of divergence type, there exists a unique PS measure class $\{\mu_{x}:x\in\mathrm{X}\}$ of dimension $\omega_{\Gamma}$ , up to scaling. In particular, the PS measure $\mu_{go}$ is the unique limit point of the following one

\mu_{go}^{s,y}=\frac{1}{\mathcal{P}_{H}(s,o,y)}\sum_{h\in H}\mathrm{e}^{-sd(go,hy)}{\mbox{Dirac}}{(hy)}

for any $y\in\mathrm{X}$ , where $\mathcal{P}_{H}(s,x,y)=\sum_{h\in H}\mathrm{e}^{-sd(x,hy)}$ is the Poincaré series associated to $H$ . Note that

(5)

\forall s>\omega_{H}:\quad\|\mu_{go}^{s,go}\|=\frac{\mathcal{P}_{H}(s,go,go)}{\mathcal{P}_{H}(s,o,go)}=\Bigg{[}\frac{\mathcal{P}_{H}(s,go,o)}{\mathcal{P}_{H^{g}}(s,o,o)}\Bigg{]}^{-1}

The proof would be finished at this point, if $H^{g}=H$ is a normal subgroup: the RHS is the inverse of the mass $\|\mu_{go}^{s,o}\|$ . The uniqueness of the limit $\mu_{go}$ concludes that $\|\mu_{go}\|=1$ , so the proof for normal subgroups is finished.

Our effort indeed comes into this stage to handle a general confined subgroup. We have to take a different routine to show (4), through an argument by contradiction. Assuming that $2\omega_{H}=\omega_{\Gamma}$ , we make a crucial use of Lemma 1.22 to obtain the following estimates on growth function of each conjugate $gHg^{-1}$

C\mathrm{e}^{n\omega_{H}}\leq\sharp(gHg^{-1}\cap A_{\Gamma}(o,n,\Delta))\leq C^{\prime}\mathrm{e}^{n\omega_{H}}

in Lemma 8.5, which yields the coarse equality of the associated Poincaré series:

\mathcal{P}_{H}(\omega_{H},o,o)\asymp\mathcal{P}_{H^{g}}(\omega_{H},o,o)

The upper bound in the above inequality uses some ingredients in proving purely exponential growth in [71]. Substituting this equation into (5) proves the boundedness of $\|\mu_{go}\|$ as above, resulting in the equality $\omega_{H}=\omega_{\Gamma}$ . This contradicts our assumption, concluding the proof of the strict inequality.

The argument by contradiction leaves open whether equality $\omega_{H}=\omega_{\Gamma}$ holds for confined subgroups of divergence type, while it is known to hold for normal subgroups.

Subgroups with nontrivial Hopf decomposition

The construction of subgroups satisfying Theorem 1.21 starts in Section 11. We first take the normal closure $G=\llangle f^{n}\rrangle$ of a contracting element $f$ for some sufficiently large $n$ . It is well known that such a $G$ is a free group of infinite rank ([24]). Removing one generator gives a subgroup $H$ of second kind, for which we show that it has the desired properties. The proof relies on the recent adaption of rotating family theory to projection complex ([17, 11]). The proof of nontrivial conservative component uses the inequality $\omega_{G}<\omega_{\Gamma}$ by the Amenability Theorem [21] for SCC action on hyperbolic spaces. We mention that the equivalence of $\omega_{G}<\omega_{\Gamma}$ with non-amenability of $\Gamma/G$ is conjectured to hold for any SCC action with contracting elements.

A guide to the sections of the paper

In the preliminary §2, we introduce necessary materials on contracting elements, convergence boundary, quasiconformal density on it, and a brief discussion on Hopf decomposition for conformal measures. The main results of the paper are then grouped into three different but closely related parts. The first part is devoted to the study of ergodic properties on boundary, establishing completely dissipative actions for subgroups with small growth (Theorem 1.10) in §4, and conservative actions for confined subgroups (Theorem 1.11) in §6. To demonstrate the main idea in our general case, we include a short section 3 to explain their proof in hyperbolic setup. The growth of confined subgroups forms the main content of the second part. Using a key lemma 5.8 obtained in the first part, we prove the shadow principle for confined subgroups in §7 and then complete the proof of strict inequality in Theorem 1.13 in §8. As a further application of the Shadow principle, Section 9 shows an inequality in Theorem 1.15 relating the growth and co-growth of confined subgroups. The last part first explains a close relation between quotient growth and Hopf decomposition and then shows the existence of nontrivial Hopf decomposition. In §10), the conservative action is characterized by slower quotient growth in Theorem 1.17. The last section 11 constructs in abundance subgroups of second kind with non-trivial Hopf decomposition (Theorem 1.21).

2. Preliminaries

Let $(\mathrm{X},d)$ be a proper geodesic metric space. Let $\mathrm{Isom}(\mathrm{X},d)$ be the isometry group endowed with compact open topology. It is well known that a subgroup $\Gamma<\mathrm{Isom}(\mathrm{X},d)$ is discrete if and only if $\Gamma$ acts properly on $\mathrm{X}$ ([59, Theorem 5.3.5]).

Let $\alpha:[s,t]\subseteq\mathbb{R}\to\mathrm{X}$ be a path parametrized by arc-length, from the initial point $\alpha^{-}:=\alpha(s)$ to the terminal point $\alpha^{+}:=\alpha(t)$ . If $[s,t]=\mathbb{R}$ , the restriction of $\alpha$ to $[a,+\infty)$ for $a\in\mathbb{R}$ is referred to as a positive ray, and its complement a negative ray. By abuse of language, we often denote them by $\alpha^{+}$ and $\alpha^{-}$ (in particular, when they represent boundary points to which the half rays converge as in Definition 2.13).

Given two parametrized points $x,y\in\alpha$ , $[x,y]_{\alpha}$ denotes the parametrized subpath of $\alpha$ going from $x$ to $y$ , while $[x,y]$ is a choice of a geodesic between $x,y\in\mathrm{X}$ .

A path $\alpha$ is called a $c$ –quasi-geodesic for $c\geq 1$ if for any rectifiable subpath $\beta$ ,

\ell(\beta)\leq c\cdot d(\beta^{-},\beta^{+})+c

where $\ell(\beta)$ denotes the length of $\beta$ .

Denote by $\alpha\cdot\beta$ (or simply $\alpha\beta$ ) the concatenation of two paths $\alpha,\beta$ provided that $\alpha^{+}=\beta^{-}$ .

Let $f,g$ be real-valued functions. Then $f\prec_{c_{i}}g$ means that there is a constant $C>0$ depending on parameters $c_{i}$ such that $f<Cg$ . The symbol $\succ_{c_{i}}$ is defined similarly, and $\asymp_{c_{i}}$ means both $\prec_{c_{i}}$ and $\succ_{c_{i}}$ are true. The constant $c_{i}$ will be omitted if it is a universal constant.

2.1. Contracting geodesics

Let $Z$ be a closed subset of $\mathrm{X}$ and $x$ be a point in $\mathrm{X}$ . By $d(x,Z)$ we mean the set-distance between $x$ and $Z$ , i.e.

d(x,Z):=\inf\big{\{}d(x,y):y\in Z\big{\}}.

Let

\pi_{Z}(x):=\big{\{}y\in Z:d(x,y)=d(x,Z)\big{\}}

be the set of closet point projections from $x$ to $Z$ . Since $X$ is a proper metric space, $\pi_{Z}(x)$ is non empty. We refer to $\pi_{Z}(x)$ as the projection set of $x$ to $Z$ . Define $\textbf{d}_{Z}(x,y):=\|\pi_{Z}(x)\cup\pi_{Z}(y)\|$ , where $\|\cdot\|$ denotes the diameter.

Definition 2.1.

We say a closed subset $Z\subseteq X$ is $C$ –contracting for a constant $C>0$ if, for all pairs of points $x,y\in X$ , we have

d(x,y)\leq d(x,Z)\quad\Longrightarrow\quad\textbf{d}_{Z}(x,y)\leq C.

Any such $C$ is called a contracting constant for $Z$ . A collection of $C$ –contracting subsets shall be referred to as a $C$ –contracting system.

An element $h\in\mathrm{Isom}(\mathrm{X})$ is called contracting if it acts co-compactly on a contracting bi-infinite quasi-geodesic. Equivalently, the map $n\in\mathbb{Z}\longmapsto h^{n}o$ is a quasi-geodesic with a contracting image.

Unless explicitly stated, let us assume from now on that $\Gamma<\mathrm{Isom}(\mathrm{X})$ is a discrete group, so $\Gamma\curvearrowright\mathrm{X}$ is a proper action (i.e. with discrete orbits and finite point stabilizers).

A group is called elementary if it is virtually $\mathbb{Z}$ or a finite group. In a discrete group, a contracting element must be of infinite order and is contained in a maximal elementary subgroup as described in the next lemma.

Lemma 2.2.

[71, Lemma 2.11] For a contracting element $h\in\Gamma$ , we have

E_{\Gamma}(h)=\{g\in\Gamma:\exists n\in\mathbb{N}_{>0},(\;gh^{n}g^{-1}=h^{n})\;\lor\;(gh^{n}g^{-1}=h^{-n})\}.

We shall suppress $\Gamma$ and write $E(h)=E_{\Gamma}(h)$ if $\Gamma$ is clear in context.

Keeping in mind the basepoint $o\in\mathrm{X}$ , the axis of $h$ is defined as the following quasi-geodesic

(6)

\mathrm{Ax}(h)=\{fo:f\in E(h)\}.

Notice that $\mathrm{Ax}(h)=\mathrm{Ax}(k)$ and $E(h)=E(k)$ for any contracting element $k\in E(h)$ .

An element $g\in\Gamma$ preserves the orientation of a bi-infinite quasi-geodesic $\gamma$ if $\alpha$ and $g\alpha$ has finite Hausdorff distance for any half ray $\alpha$ of $\gamma$ . Let $E^{+}(h)$ be the subgroup of $E(h)$ with possibly index 2 which elements preserve the orientation of their axis. Then we have

E^{+}(h)=\{g\in\Gamma:\exists 0\neq n\in\mathbb{Z},\;gh^{n}g^{-1}=h^{n}\}.

and $E^{+}(h)$ contains all contracting elements in $E(h)$ , and $E(h)\setminus E^{+}(h)$ consists of torsion elements.

Definition 2.3.

Two contracting elements $h_{1},h_{2}$ in a discrete group $\Gamma$ are called independent if the collection $\mathscr{F}=\{g\mathrm{Ax}(h_{i}):g\in\Gamma;\ i=1,2\}$ is a contracting system with bounded intersection: for any $r>0$ , there exists $L=L(r)$ so that

\forall X\neq Y\in\mathscr{F},\;\|N_{r}(X)\cap N_{r}(Y)\|\leq L.

This is equivalent to the bounded projection: $\|\pi_{X}(Y)\|\leq B$ for some $B$ independent of $X\neq Y$ .

In a possibly nondiscrete group $\Gamma$ , we say that $h_{1},h_{2}\in\Gamma$ are weakly independent if $\{h_{1}^{n}o:n\in\mathbb{Z}\}$ and $\{h_{2}^{n}o:n\in\mathbb{Z}\}$ have infinite Hausdorff distance. Note that in some papers, weak independence is referred to as independence.

Remark 2.4.

Note that two conjugate contracting elements with disjoint fixed points are weakly independent, but not independent. In the current paper, we mainly work with independent contracting elements, though many technical results hold for weakly independent contracting ones.

Definition 2.5.

Fix $r>0$ and a set $F$ in $\Gamma$ . A geodesic $\gamma$ contains an $(r,f)$ –barrier for $f\in F$ if there exists an element $g\in\Gamma$ so that

(7)

\max\{d(g\cdot o,\gamma),\;d(g\cdot fo,\gamma)\}\leq r.

By abuse of language, the point $ho$ or the axis $h\mathrm{Ax}(f)$ is called $(r,F)$ –barrier on $\gamma$ .

2.2. Extension Lemma

We fix a finite set $F\subseteq\Gamma$ of independent contracting elements and let $\mathscr{F}=\{g\mathrm{Ax}(f):f\in F,g\in\Gamma\}$ . The following notion of an admissible path allows to construct a quasi-geodesic by concatenating geodesics via $\mathscr{F}$ .

Figure 3. Admissible path

Definition 2.6 (Admissible Path).

Given $L,\tau\geq 0$ , a path $\gamma$ is called $(L,\tau)$ -admissible in $\mathrm{X}$ , if $\gamma$ is a concatenation of geodesics $p_{0}q_{1}p_{1}\cdots q_{n}p_{n}$ $(n\in\mathbb{N})$ , where the two endpoints of each $p_{i}$ lie in some $X_{i}\in\mathscr{F}$ , and the following Long Local and Bounded Projection properties hold:

(LL)

Each $p_{i}$ for $1\leq i<n$ has length bigger than $L$ , and $p_{0},p_{n}$ could be trivial;
(BP)

For each $X_{i}$ , we have $X_{i}\neq X_{i+1}$ and $\max\{\|\pi_{X_{i}}(q_{i})\|,\|\pi_{X_{i}}(q_{i+1})\|\}\leq\tau$ , where $q_{0}:=\gamma_{-}$ and $q_{n+1}:=\gamma_{+}$ by convention.

The collection $\{X_{i}:1\leq i\leq n\}$ is referred to as contracting subsets associated with the admissible path.

Remark 2.7.

(1)

The path $q_{i}$ could be allowed to be trivial, so by the (BP) condition, it suffices to check $X_{i}\neq X_{i+1}$ . It will be useful to note that admissible paths could be concatenated as follows: Let $p_{0}q_{1}p_{1}\cdots q_{n}p_{n}$ and $p_{0}^{\prime}q_{1}^{\prime}p_{1}^{\prime}\cdots q_{n}^{\prime}p_{n}^{\prime}$ be $(L,\tau)$ –admissible. If $p_{n}=p_{0}^{\prime}$ has length bigger than $L$ , then the concatenation $(p_{0}q_{1}p_{1}\cdots q_{n}p_{n})\cdot(q_{1}^{\prime}p_{1}^{\prime}\cdots q_{n}^{\prime}p_{n}^{\prime})$ has a natural $(L,\tau)$ –admissible structure.
(2)

In many situations, $\tau$ could be chosen as the bounded projection constant $B$ of $\mathscr{F}$ . In fact, if $L$ is large relative to $\tau$ , an $(L,\tau)$ –admissible path could be always truncated near contracting subsets $X_{i}$ so that it becomes an $(\hat{L},B)$ –admissible path. See [72, Lemma 2.14].

We frequently construct a path labeled by a word $(g_{1},g_{2},\cdots,g_{n})$ , which by convention means the following concatenation

[o,g_{1}o]\cdot g_{1}[o,g_{2}o]\cdots(g_{1}\cdots g_{n-1})[o,g_{n}o]

where the basepoint $o$ is understood in context. With this convention, the paths labeled by $(g_{1},g_{2},g_{3})$ and $(g_{1}g_{2},g_{3})$ respectively differ, depending on whether $[o,g_{1}o]g_{1}[o,g_{2}o]$ is a geodesic or not.

A sequence of points $x_{i}$ in a path $p$ is called linearly ordered if $x_{i+1}\in[x_{i},p^{+}]_{p}$ for each $i$ .

Definition 2.8 (Fellow travel).

Let $\gamma=p_{0}q_{1}p_{1}\cdots q_{n}p_{n}$ be an $(L,\tau)-$ admissible path. We say $\gamma$ has $r$ –fellow travel property for some $r>0$ if for any geodesic $\alpha$ with the same endpoints as $\gamma$ , there exists a sequence of linearly ordered points $z_{i},w_{i}$ ( $0\leq i\leq n$ ) on $\alpha$ such that

d(z_{i},p_{i}^{-})\leq r,\quad d(w_{i},p_{i}^{+})\leq r.

In particular, $\|N_{r}(X_{i})\cap\alpha\|\geq L$ for each $X_{i}\in\mathscr{F}(\gamma)$ .

The following result says that a local long admissible path enjoys the fellow travel property.

Proposition 2.9.

[69] For any $\tau>0$ , there exist $L,r>0$ depending only on $\tau,C$ such that any $(L,\tau)$ –admissible path has $r$ –fellow travel property. In particular, it is a $c$ –quasi-geodesic.

The next lemma gives a way to build admissible paths.

Lemma 2.10 (Extension Lemma).

For any independent contracting elements $h_{1},h_{2},h_{3}\in\Gamma$ , there exist constants $L,r,B>0$ depending only on $C$ with the following property.

Choose any element $f_{i}\in\langle h_{i}\rangle$ for each $1\leq i\leq 3$ to form the set $F$ satisfying $\|Fo\|_{\min}\geq L$ . Let $g,h\in\Gamma$ be any two elements.

(1)

There exists an element $f\in F$ such that the path

$\gamma:=[o,go]\cdot(g[o,fo])\cdot(gf[o,ho])$

is an $(L,\tau)$ –admissible path relative to $\mathscr{F}$ .
(2)

The point $go$ is an $(r,f)$ –barrier for any geodesic $[\gamma^{-},\gamma^{+}]$ .

Remark 2.11.

Since admissible paths are local conditions, we can connect via $F$ any number of elements $g\in G$ to satisfy (1) and (2). We refer the reader to [71] for a precise formulation.

The following elementary fact will be invoked frequently.

Lemma 2.12.

Assume that two words $(g_{1},f_{1},h_{1})$ and $(g_{2},f_{2},h_{2})$ label two $(L,\tau)$ –admissible paths with the same endpoints, where $(C,\tau,L)$ satisfy Proposition 2.9. For any $\Delta>1$ , there exists $R=R(\Delta,C,\tau)$ with the following property. If $|d(o,g_{1}o)-d(o,g_{2}o)|\leq\Delta$ and $d(g_{1}o,g_{2}o)>R$ , then $g_{1}=g_{2}$ .

2.3. Horofunction boundary

We recall the notion of horofunction boundary.

Fix a basepoint $o\in\mathrm{X}$ . For each $y\in\mathrm{X}$ , we define a Lipschitz map $b_{y}:\mathrm{X}\to\mathrm{X}$ by

\forall x\in\mathrm{X}:\quad b_{y}(x)=d(x,y)-d(o,y).

This family of $1$ –Lipschitz functions sits in the set of continuous functions on $\mathrm{X}$ vanishing at $o$ . Endowed with the compact-open topology, the Arzela-Ascoli Lemma implies that the closure of $\{b_{y}:y\in\mathrm{X}\}$ gives a compactification of $\mathrm{X}$ . The complement of $X$ in this compactification is called the horofunction boundary of $\mathrm{X}$ and is denoted by $\partial_{h}{\mathrm{X}}$ .

A Buseman cocycle $B_{\xi}:\mathrm{X}\times\mathrm{X}\to\mathbb{R}$ (independent of $o$ ) is given by

\forall x_{1},x_{2}\in\mathrm{X}:\quad B_{\xi}(x_{1},x_{2})=b_{\xi}(x_{1})-b_{\xi}(x_{2}).

The topological type of horofunction boundary is independent of the choice of a basepoint. Every isometry $\phi$ of $\mathrm{X}$ induces a homeomorphism on $\overline{\mathrm{X}}$ :

\forall y\in\mathrm{X}:\quad\phi(\xi)(y):=b_{\xi}(\phi^{-1}(y))-b_{\xi}(\phi^{-1}(o)).

Depending on the context, we may use both $\xi$ and $b_{\xi}$ to denote a point in the horofunction boundary.

Finite difference relation.

Two horofunctions $b_{\xi},b_{\eta}$ have $K$ –finite difference for $K\geq 0$ if the $L^{\infty}$ –norm of their difference is $K$ –bounded:

\|b_{\xi}-b_{\eta}\|_{\infty}\leq K.

The locus of $b_{\xi}$ consists of horofunctions $b_{\eta}$ so that $b_{\xi},b_{\eta}$ have $K$ –finite difference for some $K>0$ . The loci $[b_{\xi}]$ of horofunctions $b_{\xi}$ form a finite difference equivalence relation $[\cdot]$ on $\partial_{h}{\mathrm{X}}$ . The locus $[\Lambda]$ of a subset $\Lambda\subseteq\partial_{h}{\mathrm{X}}$ is the union of loci of all points in $\Lambda$ .

If $x_{n}\in\mathrm{X}\to\xi\in\partial{\mathrm{X}}$ and $y_{n}\in\mathrm{X}\to\eta\in\partial{\mathrm{X}}$ are sequences with $\sup_{n\geq 1}d(x_{n},y_{n})<\infty$ , then $[\xi]=[\eta]$ .

2.4. Convergence boundary

Let $(\mathrm{X},d)$ be a proper metric space admitting an isometric action of a non-elementary countable group $\Gamma$ with a contracting element. Consider a metrizable compactification $\overline{\mathrm{X}}:=\partial{\mathrm{X}}\cup\mathrm{X}$ , so that $\mathrm{X}$ is open and dense in $\overline{\mathrm{X}}$ . We also assume that the action of $\mathrm{Isom}(\mathrm{X})$ extends by homeomorphism to $\partial{\mathrm{X}}$ .

We equip $\partial{\mathrm{X}}$ with a $\mathrm{Isom}(\mathrm{X})$ –invariant partition $[\cdot]$ : $[\xi]=[\eta]$ implies $[g\xi]=[g\eta]$ for any $g\in\mathrm{Isom}(\mathrm{X})$ . We say that $\xi$ is minimal if $[\xi]=\{\xi\}$ , and a subset $U$ is $[\cdot]$ –saturated if $U=[U]$ .

We say that $[\cdot]$ restricts to be a closed partition on a $[\cdot]$ –saturated subset $U\subseteq\partial{\mathrm{X}}$ if $x_{n}\in U\to\xi\in\partial{\mathrm{X}}$ and $y_{n}\in U\to\eta\in\partial{\mathrm{X}}$ are two sequences with $[x_{n}]=[y_{n}]$ , then $[\xi]=[\eta]$ . (The points $\xi,\eta$ are not necessarily in $U$ .) If $U=\partial{\mathrm{X}}$ , this is equivalent to saying that the relation $\{(\xi,\eta):[\xi]=[\eta]\}$ is a closed subset in $\partial{\mathrm{X}}\times\partial{\mathrm{X}}$ , so the quotient space $[\partial{\mathrm{X}}]$ is Hausdorff. In general, $[\cdot]$ may not be closed over the whole $\partial{\mathrm{X}}$ (e.g., the horofunction boundary with finite difference relation), but is closed when restricted to certain interesting subsets; see for example Assumption (C) below.

We say that $x_{n}$ tends (resp. accumulates) to $[\xi]$ if the limit point (resp. any accumulate point) is contained in the subset $[\xi]$ . This implies that $[x_{n}]$ tends or accumulates to $[\xi]$ in the quotient space $[\Lambda(\Gamma o)]$ . So, an infinite ray $\gamma$ terminates at $[\xi]\in\partial{\mathrm{X}}$ if any sequence of points in $\gamma$ accumulates in $[\xi]$ .

Figure 4. Assumptions (A)(B)(C) in convergence boundary

Recall that $\Omega_{x}(A):=\{y\in\mathrm{X}:\exists[x,y]\cap A\neq\emptyset\}$ is the cone of a subset $A\subseteq\mathrm{X}$ with light source at $x$ . A sequence of subsets $A_{n}\subseteq\mathrm{X}$ is escaping if $d(o,A_{n})\to\infty$ for some (or any) $o\in\mathrm{X}$ .

Definition 2.13.

We say that $(\overline{\mathrm{X}},[\cdot])$ is a convergence compactification of $\mathrm{X}$ if the following assumptions hold.

(A)

Any contracting geodesic ray $\gamma$ accumulates into a closed subset $[\xi]$ for some $\xi\in\partial{\mathrm{X}}$ ; and any sequence $y_{n}\in\mathrm{X}$ with escaping projections $\pi_{\gamma}(y_{n})$ tends to $[\xi]$ .
(B)

Let $\{X_{n}\subseteq\mathrm{X}:n\geq 1\}$ be an escaping sequence of $C$ –contracting quasi-geodesics for some $C>0$ . Then for any given $o\in\mathrm{X}$ , there exists a subsequence of $\{Y_{n}:=\Omega_{o}(X_{n}):n\geq 1\}$ (still denoted by $Y_{n}$ ) and $\xi\in\partial{\mathrm{X}}$ such that $Y_{n}$ accumulates to $[\xi]$ : any convergent sequence of points $y_{n}\in Y_{n}$ tends to a point in $[\xi]$ .
(C)

The set $\mathcal{C}$ of non-pinched points $\xi\in\partial{\mathrm{X}}$ is non-empty. We say $\xi$ is non-pinched if $x_{n},y_{n}\in\mathrm{X}$ are two sequences of points converging to $[\xi]$ , then $[x_{n},y_{n}]$ is an escaping sequence of geodesic segments. Moreover, for any $\xi\in\mathcal{C}$ , the partition $[\cdot]$ restricted to $G[\xi]$ forms a closed relation.

Remark 2.14.

Assumption (C) is a non-triviality condition: any compactification with the coarsest partition asserting $\partial{\mathrm{X}}$ as one $[\cdot]$ –class satisfy (A) and (B). We require $\mathcal{C}$ to be non-empty to exclude such situations. The “moreover” statement is newly added relative to the one in [72]. If the restriction of the partition to $\mathcal{C}$ is maximal in the sense that every $[\cdot]$ –class is singleton, then the “moreover” assumption is always true.

By Assumption (A), a contracting quasi-geodesic ray $\gamma$ determines a unique $[\cdot]$ –class of a boundary point denoted by $[\gamma^{+}]$ . Similarly, the positive and negative rays of a bi-infinite contracting quasi-geodesic $\gamma:(-\infty,\infty)\to\mathrm{X}$ determine respectively two boundary $[\cdot]$ –classes denoted by $[\gamma^{+}]$ and $[\gamma^{-}]$ .

Examples 2.15.

The first three convergence boundaries below are equipped with a maximal partition $[\cdot]$ (that is, $[\cdot]$ –classes are singletons).

(1)

Hyperbolic space $\mathrm{X}$ with Gromov boundary $\partial{\mathrm{X}}$ , where all boundary points are non-pinched.
(2)

CAT(0) space $\mathrm{X}$ with visual boundary $\partial{\mathrm{X}}$ (homeomorphic to horofunction boundary), where all boundary points are non-pinched.
(3)

The Cayley graph $\mathrm{X}$ of a relatively hyperbolic group with Bowditch or Floyd boundary $\partial{\mathrm{X}}$ , where conical limit points are non-pinched.

If $\mathrm{X}$ is infinitely ended, we could also take $\partial{\mathrm{X}}$ as the end boundary with the same statement.
(4)

Teichmüller space $\mathrm{X}$ with Thurston boundary $\partial{\mathrm{X}}$ , where $[\cdot]$ is given by Kaimanovich-Masur partition [41] and uniquely ergodic points are non-pinched.
(5)

Any proper metric space $\mathrm{X}$ with horofunction boundary $\partial{\mathrm{X}}$ , where $[\cdot]$ is given by finite difference partition and all boundary points are non-pinched. If $\mathrm{X}$ is the cubical CAT(0) space, the horofunction boundary is exactly the Roller boundary. If $\mathrm{X}$ is the Teichmüller space with Teichmüller metric, the horofunction boundary is the Gardiner-Masur boundary ([47, 68]).

From these examples, one can see that the convergence boundary is not a canonical object associated to a given proper metric space. In some sense, the horofunction boundary provides a “universal” convergence boundary with non-pinched points for any proper action.

Theorem 2.16.

[72, Theorem 1.1] The horofunction boundary is a convergence boundary with finite difference relation $[\cdot]$ , where all boundary points are non-pinched.

Proof.

Only the “moreover” statement in Assumption (C) requires clarification. Under the finite difference relation $[\cdot]$ , if $[\xi]$ is $K$ –bounded for some $K>0$ , then any $g[\xi]$ is $K$ –bounded as well. According to the topology of the horofunction compactification, if $|b_{x_{n}}(\cdot)-b_{y_{n}}(\cdot)|\leq K$ and $x_{n}\to\xi,y_{n}\to\eta$ , then $|b_{\xi}(\cdot)-b_{\eta}(\cdot)|\leq K$ . This shows $[\xi]=[\eta]$ (however, $[\xi]$ may not be $K$ –bounded). ∎

If $f\in\mathrm{Isom}(\mathrm{X})$ is a contracting element, the positive and negative rays of a contracting quasi-geodesic $n\in\mathbb{Z}\mapsto f^{n}o$ determine two boundary points denoted by $[f^{+}]$ and $[f^{-}]$ respectively. Write $[f^{\pm}]=[f^{+}]\cup[f^{-}]$ . We say that $f$ is non-pinched if $[f^{\pm}]\subseteq\mathcal{C}$ are non-pinched. On the horofunction boundary, any contracting element is non-pinched by Theorem 2.16. In contrast, there exist pinched contracting elements in the Bowditch boundary of a relatively hyperbolic group; for instance, those contracting elements in a parabolic group are pinched. The following two results fail if the non-pinched condition is dropped.

Recall that $\Gamma<\mathrm{Isom}(\mathrm{X})$ is a discrete group, which we assume in the next results.

Lemma 2.17.

If $f\in\Gamma$ is a non-pinched contracting element, then $E(f)$ coincides with the set stabilizer of the two fixed points $[f^{\pm}]$ .

Proof.

By Assumption A, the two half ray of the axis $\mathrm{Ax}(f)$ accumulates either to $[f^{-}]$ or to $[f^{+}]$ . The same holds for $g\mathrm{Ax}(f)$ , which accumulates to $g[f^{\pm}]$ . By definition, $E(f)$ is the set of elements $g\in G$ which preserves $\mathrm{Ax}(f)$ up to finite Hausdorff distance. Let $g\in G$ preserve $[f^{\pm}]$ but $g\notin E(f)$ . Choose two sequence of points $x_{n},y_{n}\in g\mathrm{Ax}(f)$ so that $x_{n}\to[f^{-}],y_{n}\in[f^{+}]$ , and let $z_{n}\in\pi_{\mathrm{Ax}(f)}(x_{n})$ and $w_{n}\in\pi_{\mathrm{Ax}(f)}(y_{n})$ . We claim that both $z_{n}$ and $w_{n}$ are unbounded sequences. If not, assume for concreteness that $z_{n}$ is bounded. Choose $u_{n}\in\mathrm{Ax}(f)$ tending to $[f^{-}]$ ( $[f^{-}]$ , if $w_{n}$ is bounded). The contracting property implies $[x_{n},u_{n}]\cap B(z_{n},2C)\neq\emptyset$ , where $C$ is the contracting constant of $\mathrm{Ax}(f)$ . As $z_{n}$ is bounded and $x_{n},u_{n}$ tend to $[f^{-}]$ , this contradicts $[f^{-}]\subseteq\mathcal{C}$ by Definition 2.13(C). ∎

Lemma 2.18.

[72, Lemma 3.12] If $f,g\in\Gamma$ are two non-pinched independent contracting elements, then either $[f^{\pm}]=[g^{\pm}]$ or $[f^{\pm}]\cap[g^{\pm}]=\emptyset$ .

Lemma 2.19.

[72, Lemma 3.10] Let $g,f\in\Gamma$ be two independent contracting elements. Then for all $n\gg 0$ , $h_{n}:=g^{n}f^{n}$ is a contracting element so that $[h_{n}^{+}]$ tends to $[g^{+}]$ and $[h_{n}^{-}]$ tends to $[f^{-}]$ .

The following lemma is often used to choose an independent contracting element.

Lemma 2.20.

Let $H<\mathrm{Isom}(\mathrm{X})$ be a non-elementary discrete group. Assume that $f\in H$ is a contracting element. Then there exists a contracting element $h\in H$ that is independent with $f$ .

Proof.

The set of translated axis $\mathscr{F}:=\{g\mathrm{Ax}(f):g\in\Gamma\}$ has bounded intersection. Pick two distinct conjugates denoted by $f_{1},f_{2}$ of $f$ . By Lemma 2.19, $h:=f_{1}^{n}f_{2}^{n}$ is a contracting element whose fixed points tend to $[f_{1}^{+}]$ and $[f_{2}^{-}]$ respectively for any large $n\gg 0$ . Note that the axis $\mathrm{Ax}(h)$ of $h$ has an overlap with $\mathrm{Ax}(f_{1})$ and $\mathrm{Ax}(f_{2})$ of a large length $d(o,f_{i}^{n}o)$ for $i=1,2$ . Taking into account every pair of elements in $\mathscr{F}$ has bounded intersection, we conclude that $\mathrm{Ax}(h)$ has bounded intersection with every element in $\mathscr{F}$ . That is, $h$ is independent with $f$ by definition. ∎

Let $U$ be a subset of $\mathrm{X}$ . The limit set of $U$ , denoted by $\Lambda(U)$ , is the set of all accumulation points of $U$ in $\partial{\mathrm{X}}$ . If $H<\Gamma$ is a subgroup, $\Lambda(Ho)$ may depend on the choice of the basepoint $o\in\mathrm{X}$ , but the $[\cdot]$ –loci $[\Lambda(Ho)]$ does not by Definition 2.13(B). For this reason, we shall refer $[\Lambda(Ho)]$ as the limit set of a subgroup $H$ . Moreover, the limit set of a discrete group $\Gamma$ enjoys the following desirable property as in the general theory of convergence group action.

Lemma 2.21.

[72, Lemma 3.9] Assume that $\Gamma$ is non-elementary. Let $f\in\Gamma$ be a contracting element. Then $[\Lambda(\Gamma o)]=[\overline{\Gamma\xi}]$ for any $\xi\in[f^{\pm}]$ .

2.5. Quasi-conformal density and Patterson’s construction

Let $\overline{\mathrm{X}}:=\partial{\mathrm{X}}\cup\mathrm{X}$ be a convergence compactification, with a nonempty set $\mathcal{C}$ of non-pinched points in Definition 2.13. We need to restrict to a smaller subset of $\mathcal{C}$ , on which some weak continuity of Busemann cocycles can be guaranteed.

Assumption D.

There exists a subset $\mathcal{C}^{\mathrm{hor}}\subseteq\mathcal{C}$ with a family of Buseman quasi-cocycles

\big{\{}B_{\xi}:\quad\mathrm{X}\times\mathrm{X}\longrightarrow\mathbb{R}\big{\}}_{\xi\in\mathcal{C}^{\mathrm{hor}}}

so that for any $x,y\in\mathrm{X}$ , we have

(8)

\displaystyle\limsup_{z\to[\xi]}|B_{\xi}(x,y)-B_{z}(x,y)|\leq\epsilon,

where $\epsilon\geq 0$ may depend on $[\xi]$ but not on $x,y$ .

For a given $\epsilon$ , let $\mathcal{C}^{\mathrm{hor}}_{\epsilon}$ denote the set of points $\xi\in\mathcal{C}^{\mathrm{hor}}$ for which (8) holds. Thus, $\mathcal{C}^{\mathrm{hor}}=\bigcup_{\epsilon\geq 0}\mathcal{C}^{\mathrm{hor}}_{\epsilon}$ .

In practice, the assumption (8) could allow to take the following concrete definition:

\displaystyle B_{\xi}(x,y)

\displaystyle=\limsup_{z_{n}\to\xi}\left[d(x,z_{n})-d(y,z_{n})\right]

The following facts shall be used implicitly.

(9)

\begin{array}[]{l}B_{\xi}(x,y)\leq d(x,y)\\ B_{\xi}(x,y)=B_{g\xi}(gx,gy),\;\forall g\in\mathrm{Isom}(\mathrm{X})\end{array}

Horofunctions in convergence boundary

We now define Buseman quasi-cocycles at a $[\cdot]$ –class $[\xi]$ . Namely, given $\xi\in\mathcal{C}^{\mathrm{hor}}_{\epsilon}$ , define a Busemann quasi-cocycle at $\xi$ (resp. $[\xi]$ ) as follows

\displaystyle B_{[\xi]}(x,y)=\limsup_{z_{n}\to[\xi]}\left[d(x,z_{n})-d(y,z_{n})\right]

where the convergence $z_{n}\in\mathrm{X}\to\xi$ (resp. $z_{n}\to[\xi]$ ) takes place in $\overline{\mathrm{X}}=\mathrm{X}\cup\partial{\mathrm{X}}$ . The equations in (9) hold for $B_{[\xi]}(x,y)$ , and moreover, for all $\xi\in[\xi]$ ,

(10)

\begin{array}[]{rl}|B_{\xi}(x,y)-B_{[\xi]}(x,y)|&\leq\epsilon\\ |B_{[\xi]}(x,y)-B_{[\xi]}(x,z)-B_{[\xi]}(z,y)|&\leq 2\epsilon\end{array}

Horoballs at $[\cdot]$ –classes

We give the following tentative definition of horoballs at a $[\cdot]$ –class in a general convergence boundary. Given a real number $L$ , we define the horoball of (algebraic) depth $L$ . We take the convention that the center $\xi$ has depth $-\infty$ .

Definition 2.22.

Let $\xi\in\mathcal{C}^{\mathrm{hor}}_{\epsilon}$ for $\epsilon\geq 0$ . We define a horoball centered at $[\xi]$ of depth $L$ as follows

\mathcal{HB}([\xi],L)=\{y\in\mathrm{X}:B_{[\xi]}(y,o)\leq L\}

We omit $L$ if $L=0$ or it does not matter in context.

As the limit supper of continuous functions, $B_{[\xi]}(x,y)$ is lower semi-continuous, so $\mathcal{HB}([\xi],L)$ is a closed subset. By abuse of language, we say the level set $\{y\in\mathrm{X}:B_{[\xi]}(y,o)=L\}$ is a horosphere or the boundary of $\mathcal{HB}([\xi],L)$ .

Lemma 2.23.

For $L_{1}>L_{2}$ , we have

(1)

$\mathcal{HB}([\xi],L_{1})\subseteq N_{L}(\mathcal{HB}([\xi],L_{2}))$ for $L=L_{1}-L_{2}$ .
(2)

$\mathcal{HB}([\xi],L_{2})$ has distance at least $L-2\epsilon$ to the complement $\mathrm{X}\setminus\mathcal{HB}([\xi],L_{1})$ .

Proof.

(1). Observe that, for any $x\in\mathrm{X}$ and $\xi\in\partial{\mathrm{X}}$ , there exists a geodesic ray $\gamma$ starting at $x$ so that $B_{\xi}(x,y)=d(x,y)$ for any $y\in\gamma$ . ( $\gamma$ is called a gradient line in some literature). By an Ascoli-Arzela argument, $\gamma$ is obtained as a limiting ray of $[x,z_{n}]$ when $z_{n}\to\xi$ . Thus, any point $x\in\mathcal{HB}([\xi],L_{1})$ has a distance $L$ to some $y\in\mathcal{HB}([\xi],L_{2})$ .

(2). If $d(x,y)\leq L-2\epsilon$ for some $x\in\mathcal{HB}([\xi],L_{2})$ and $y\in\mathrm{X}$ , we obtain $B_{[\xi]}(y,o)\leq B_{[\xi]}(y,x)+B_{[\xi]}(x,o)+2\epsilon\leq L+L_{2}\leq L_{1}$ so $y\in\mathcal{HB}([\xi],L_{1})$ . The conclusion follows. ∎

By abuse of language, we also call the set $\mathcal{HB}_{L}([\xi],x)$ a horoball in any $\eta\in[\xi]$ . In other words, all points in the same class $[\xi]$ share a common horoball.

Let $\mathcal{M}^{+}(\overline{\mathrm{X}})$ be the set of finite positive Radon measures on $\overline{\mathrm{X}}:=\partial{\mathrm{X}}\cup\mathrm{X}$ , on which $\Gamma$ acts by push-forward: for any Borel set $A$ ,

g_{\star}\mu(A)=\mu(g^{-1}A).

Definition 2.24.

Let $\omega\in[0,\infty[$ . A map

	$\displaystyle\mu:\quad\mathrm{X}\quad\longrightarrow$	$\displaystyle\quad\mathcal{M}^{+}(\overline{\mathrm{X}})$
	$\displaystyle x\quad\longmapsto$	$\displaystyle\quad\mu_{x}$

is a $\omega$ –dimensional, $\Gamma$ –quasi-equivariant, quasi-conformal density if for any $g\in\Gamma$ and $x,y\in\mathrm{X}$ ,

(11)		$\displaystyle\mu_{gx}-\mathrm{a.e.}\;\xi\in\partial{\mathrm{X}}:$	$\displaystyle\quad\frac{dg_{\star}\mu_{x}}{d\mu_{gx}}(\xi)\in[\frac{1}{\lambda},\lambda],$
(12)		$\displaystyle\mu_{y}-\mathrm{a.e.}\;\xi\in\partial{\mathrm{X}}:$	$\displaystyle\quad\frac{1}{\lambda}e^{-\omega B_{\xi}(x,y)}\leq\frac{d\mu_{x}}{d\mu_{y}}(\xi)\leq\lambda e^{-\omega B_{\xi}(x,y)}$

for a universal constant $\lambda\geq 1$ . We normalize $\mu_{o}$ to be a probability: its mass $\|\mu_{o}\|=\mu_{o}(\overline{\mathrm{X}})=1$ .

If $\{\mu_{x}:x\in\mathrm{X}\}$ is supported on non-pinched boundary points (i.e.: $\mu_{o}(\mathcal{C})=1$ ), we say it is a non-trivial quasi-conformal density.

Remark 2.25.

A non-trivial quasi-conformal density is much weaker than saying that $\mu_{o}$ is supported on the conical points. Take for instance the horofunction boundary $\partial{\mathrm{X}}$ , where $\mathcal{C}=\partial{\mathrm{X}}$ is the whole boundary. Moreover, all Patterson-Sullivan measures are non-trivial quasi-conformal density in Examples 1.9.

If $\lambda=1$ for (11), the map $\mu:\mathrm{X}\to\mathcal{M}^{+}(\overline{\mathrm{X}})$ is $\Gamma$ –equivariant: that is, $\mu_{gx}=g_{\star}\mu_{x}$ (equivalently, $\mu_{gx}(gA)=\mu_{x}(A)$ ). If in both (11) and (12), $\lambda=1$ , we call $\mu$ a conformal density.

Patterson’s construction of quasi-conformal density

Fix a basepoint $o\in\mathrm{X}$ . Consider the orbital points in the ball of radius $R>0$ :

(13)

N_{\Gamma}(o,n)=\{v\in\Gamma o:d(o,v)\leq n\}

The critical exponent $\omega_{A}$ for a subset $A\subseteq\Gamma$ is independent of the choice of $o\in\mathrm{X}$ :

(14)

\omega_{A}=\limsup\limits_{R\to\infty}\frac{\log\sharp(N_{\Gamma}(o,R)\cap Ao)}{R},

which is intimately related to the (partial) Poincaré series

(15)

s\geq 0,x,y\in\mathrm{X},\quad\mathcal{P}_{A}(s,x,y)=\sum\limits_{g\in A}e^{-sd(x,gy)}

as $\mathcal{P}_{A}(s,x,y)$ diverges for $s<\omega_{A}$ and converges for $s>\omega_{A}$ . The action $\Gamma\curvearrowright\mathrm{X}$ is called of divergence type (resp. convergence type) if $\mathcal{P}_{\Gamma}(s,x,y)$ is divergent (resp. convergent) at $s=\omega_{\Gamma}$ .

Fix $\Delta\geq 1$ . The family of the annulus-like sets of radius $n$ centered at $o\in\mathrm{X}$

(16)

A_{\Gamma}(o,n,\Delta)=\{v\in\Gamma o:|d(o,v)-n|\leq\Delta\}

covers $Go$ with multiplicity at most $2\Delta$ . It is useful to keep in mind that for $s>\omega_{A},x,y\in\mathrm{X},$

(17)

\quad\mathcal{P}_{A}(s,o,o)\asymp_{\Delta}\sum\limits_{g\in A\cap A_{\Gamma}(o,n,\Delta)}e^{-sd(o,go)}

Fix $y\in\mathrm{X}$ . We start by constructing a family of measures $\{\mu_{x}^{s,y}\}_{x\in\mathrm{X}}$ supported on $\Gamma y$ for any given $s>\omega_{\Gamma}$ . Assume that $\mathcal{P}_{\Gamma}(s,x,y)$ is divergent at $s=\omega_{\Gamma}$ . Set

(18)

\mu_{x}^{s,y}=\frac{1}{\mathcal{P}_{\Gamma}(s,o,y)}\sum\limits_{g\in\Gamma}e^{-sd(x,gy)}\cdot{\mbox{Dirac}}{(gy)},

where $s>\omega_{\Gamma}$ and $x\in\mathrm{X}$ . Note that $\mu^{s,y}_{o}$ is a probability measure supported on $\Gamma y$ . If $\mathcal{P}_{\Gamma}(s,x,y)$ is convergent at $s=\omega_{\Gamma}$ , the Poincaré series in (18) needs to be replaced by a modified series as in [54].

Choose $s_{i}\to\omega_{\Gamma}$ such that for each $x\in\mathrm{X}$ , $\mu_{x}^{s_{i},y}$ is a convergent sequence in $\mathcal{M}(\Lambda(\Gamma o))$ . The family of limit measures $\mu_{x}^{y}=\lim\mu_{x}^{s_{i},y}$ ( $x\in\mathrm{X}$ ) are called Patterson-Sullivan measures.

By construction, Patterson-Sullivan measures are by no means unique a priori, depending on the choice of $y\in\mathrm{X}$ and $s_{i}\to\omega_{\Gamma}$ . Note that $\mu_{o}^{y}(\Lambda(\Gamma o))=1$ is a normalized condition, where $o\in\mathrm{X}$ is a priori chosen basepoint. In what follows, we usually write $\mu_{x}=\mu_{x}^{o}$ for $x\in\mathrm{X}$ (i.e.: $y=o$ ).

There are other sources of conformal densities, not necessarily coming from Patterson’s construction. Here we describe in some details the construction of conformal densities on Thurston boundary. See [38] for other examples constructed on ends of hyperbolic 3-manifolds.

Example 2.26.

Consider the Teichmüller space $\mathcal{T}_{g}$ of a closed oriented surface $\Sigma_{g}$ ( $g\geq 2$ ). The space $\mathscr{MF}$ of measured foliations on $\Sigma_{g}$ , which is homeomorphic to $\mathbb{R}^{6g-6}$ , admits a $\mathrm{Mod}(\Sigma_{g})$ –invariant ergodic measure, $\mu_{\mathrm{Th}}$ , called Thurston measure by the work of Masur and Veech. Positive reals $\mathbb{R}_{>0}$ act on $\mathscr{MF}$ by scaling, and let

\pi:\mathscr{MF}\longmapsto\mathscr{PMF}=\mathscr{MF}/\mathbb{R}_{>0}

be the natural quotient map.

$\mathscr{PMF}$ is also called the Thurston boundary, and it gives a convergence boundary for $\mathcal{T}_{g}$ with equipped with the Teichmüller metric. Here we take the Kaimanovich-Masur partition $[\cdot]$ , whose precise description is not relevant here, but the set $\mathscr{UE}$ of uniquely ergodic points in $\mathscr{PMF}$ is partitioned into singletons. It is a well-known fact that $\mu_{\mathrm{Th}}$ is supported on $\pi^{-1}(\mathscr{UE})$ .

The $\mu_{\mathrm{Th}}$ induces a $(6g-6)$ –dimensional, $\mathrm{Mod}(\Sigma_{g})$ –equivariant, conformal density on $\mathscr{PMF}$ as follows. Given $x\in\mathcal{T}_{g}$ , consider the extremal length function

\mathrm{Ext}_{x}:\mathscr{MF}\longmapsto\mathbb{R}_{\geq 0}

which is square homogeneous, $\mathrm{Ext}_{x}(t\xi)=t^{2}\mathrm{Ext}_{x}(\xi)$ . Take the ball-like set $B_{\mathrm{Ext}}(x)=\{\xi\in\mathscr{MF}:\mathrm{Ext}_{x}(\xi)\leq 1\}.$ For any $A\subseteq\mathscr{PMF}$ , set $\mu_{x}(A)=\mu_{\mathrm{Th}}(B_{\mathrm{Ext}}(x)\cap\pi^{-1}{A})$ . Direct computation gives

\mu_{y}-\mathrm{a.e.}\;\xi\in\mathscr{PMF}:\;\frac{d\mu_{x}}{d\mu_{y}}(\xi)=\left[\frac{\sqrt{\mathrm{Ext}_{x}(\xi)}}{\sqrt{\mathrm{Ext}_{y}(\xi)}}\right]^{6g-6}

The family $\{\mu_{x}\}$ forms a conformal density of dimension $6g-6$ : the term in the right-hand bracket coincides with $\mathrm{e}^{-B_{\xi}(x,y)}$ for $\xi\in\mathscr{UE}$ , and $\mu_{x}$ charges the full measure to $\mathscr{UE}$ . Note that this family is the same as the conformal density obtained from the action $\mathrm{Mod}(\Sigma_{g})\curvearrowright\mathcal{T}_{g}$ through the Patterson construction above (see [72]).

Theorem 2.27.

Suppose that $G$ acts properly on a proper geodesic space $\mathrm{X}$ compactified with horofunction boundary $\partial_{h}{\mathrm{X}}$ . Then the family $\{\mu_{x}:x\in\mathrm{X}\}$ of Patterson-Sullivan measures is a $\omega_{G}$ -dimensional $G$ -equivariant conformal density supported on $[\Lambda Go]$ .

In the sequel, we write PS-measures as shorthand for Patterson-Sullivan measures.

Let $\{\mu_{x}:x\in\mathrm{X}\}$ be a nontrivial $\omega$ –dimensional $\Gamma$ –equivariant quasiconformal density on a convergence compactification $\mathrm{X}\cup\partial{\mathrm{X}}$ : $\mu_{o}$ is supported on the set $\mathcal{C}$ of non-pinched points in Definition 2.13(C).

2.6. Shadow Principle

Let $F\subseteq\Gamma$ be a set of three (mutually) independent contracting elements $f_{i}$ ( $i=1,2,3$ ), which form a $C$ -contracting system

(19)

\mathscr{F}=\{g\cdot\mathrm{Ax}(f_{i}):g\in\Gamma\}

where the axis $\mathrm{Ax}(f_{i})$ in (6) is $C$ -contracting with $C$ depending on the choice of the basepoint $o\in\mathrm{X}$ . We may often assume $d(o,fo)$ is large as possible, by taking sufficiently high power of $f\in F$ . The contracting constant $C$ is not effected.

First of all, define the usual cone and shadow:

\Omega_{x}(y,r):=\{z\in\mathrm{X}:\exists[x,z]\cap B(y,r)\neq\emptyset\}

and $\Pi_{x}(y,r)\subseteq\partial{\mathrm{X}}$ be the topological closure in $\partial{\mathrm{X}}$ of $\Omega_{x}(y,r)$ .

The partial shadows $\Pi_{o}^{F}(go,r)$ and cones $\Omega_{o}^{F}(go,r)$ defined in the following depend on the choice of a contracting system $\mathscr{F}$ in (19).

Definition 2.28 (Partial cone and shadow).

For $x\in\mathrm{X},y\in\Gamma o$ , the $(r,F)$ –cone $\Omega_{x}^{F}(y,r)$ is the set of elements $z\in\mathrm{X}$ such that $y$ is a $(r,F)$ –barrier for some geodesic $[x,z]$ .

The $(r,F)$ –shadow $\Pi_{x}^{F}(y,r)\subseteq\partial{\mathrm{X}}$ is the topological closure in $\partial{\mathrm{X}}$ of the cone $\Omega_{x}^{F}(y,r)$ .

The follow terminology is from Roblin [60].

Definition 2.29.

We say that $\{\mu_{x}:x\in\mathrm{X}\}$ satisfies Shadow Principle over a subset $Z\subseteq\mathrm{X}$ if there is some large constant $r_{0}>0$ so that the following holds

\displaystyle\|\mu_{y}\|\mathrm{e}^{-\omega\cdot d(x,y)}\prec\mu_{x}(\Pi_{x}(y,r))\prec_{r}\|\mu_{y}\|\mathrm{e}^{-\omega\cdot d(x,y)}

for any $x,y\in Z$ and $r>r_{0}$ , where the implied constant depends on $Z$ .

The most fundamental example is provided by the Sullivan’s Shadow Lemma, where $Z$ could be taken to be any $\Gamma$ –cocompact subset (by $\Gamma$ –equivariance, $\|\mu_{x}\|$ thus remains uniformly bounded over $x\in Z$ ). Here are some other examples.

Examples 2.30.

(1)

Roblin realized that $Z$ could be enlarged to $Z:=N(\Gamma)o$ , provided that the normalizer $N(\Gamma)$ is a discrete subgroup.
(2)

Let $\mathrm{X}$ be a rank-1 symmetric space. If $\omega$ is the Hausdorff dimension of the visual boundary $\partial{\mathrm{X}}$ , there exists a unique (up to scaling) $\omega$ –dimensional $\Gamma$ –equivariant conformal density $\{\mu_{x}:x\in\mathrm{X}\}$ on $\partial{\mathrm{X}}$ . It satisfies the Shadow Principle over the whole space $Z=\mathrm{X}$ , where $\|\mu_{x}\|=1$ for any $x\in\mathrm{X}$ .
(3)

We shall establish the Shadow Principle for confined subgroups in §7.
(4)

We conjecture that the unique $(6g-6)$ –dimensional conformal density on $\mathscr{PMF}$ also satisfies the Shadow Principle over $Z=\mathcal{T}_{g}$ . See the relevance to Question 1.18 as explained in Remark 10.12.

We now recall the following shadow lemma on the convergence boundary.

Lemma 2.31 ([72, Lemma 6.3]).

Let $\{\mu_{x}\}_{x\in\mathrm{X}}$ be a nontrivial $\omega$ –dimensional $G$ –equivariant quasi-conformal density for some $\omega>0$ (i.e. supported on the set $\mathcal{C}\subseteq\partial{\mathrm{X}}$ of non-pinched points). Then there exist $r_{0},L_{0}>0$ with the following property.

Assume that $d(o,fo)>L_{0}$ for each $f\in F$ . For given $r\geq r_{0}$ , there exist $C_{0}=C_{0}(F),C_{1}=C_{1}(F,r)$ such that

\begin{array}[]{rl}C_{0}\mathrm{e}^{-\omega\cdot d(o,go)}\leq\mu_{o}(\Pi_{o}^{F}(go,r))\leq\mu_{o}(\Pi_{o}(go,r))\leq C_{1}\mathrm{e}^{-\omega\cdot d(o,go)}\end{array}

for any $go\in Go$ .

Remark 2.32.

We emphasize that the $\omega$ –dimensional $\Gamma$ –conformal density exists only for $\omega\geq\omega_{\Gamma}$ ([72, Prop. 6.6]). Even if the action $\Gamma\curvearrowright\mathrm{X}$ is of divergence action (i.e. a non-uniform lattice in $\mathrm{Isom}(\mathbb{H}^{n})$ ), a $\omega$ –dimensional conformal density may exist for $\omega>\omega_{\Gamma}$ . See [38] for relevant discussions.

In what follows, when speaking of $\Pi_{o}^{F}(go,r)$ , we assume $r,F$ satisfy the Shadow Lemma 2.31.

2.7. Conical points

We now give the definition of a conical point. Recall that $\mathcal{C}\subseteq\partial{\mathrm{X}}$ is the set of non-pinched boundary points in Definition 2.13. Roughly speaking, conical points are non-pinched and shadowed by infinitely many contracting segments with fixed parameter.

Definition 2.33.

A point $\xi\in\mathcal{C}$ is called $(r,F)$ –conical point if for some $x\in\Gamma o$ , the point $\xi$ lies in infinitely many $(r,F)$ –shadows $\Pi_{x}^{F}(y_{n},r)$ for $y_{n}\in\Gamma o$ . We denote by $\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ the set of $(r,F)$ –conical points.

We also denote by $\Lambda^{\textrm{con}}{(\Gamma o)}$ the set of conical points $\xi\in\mathcal{C}$ for which there exists $g_{n}o\in\Gamma o$ satisfying $\xi\in\Pi_{o}(g_{n}o,r)$ for some large $r>0$ . By definition, $\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}\subseteq\Lambda^{\textrm{con}}{(\Gamma o)}$ .

Figure 5. Conical points

The following is the Borel-Cantelli Lemma stated in terms of conical points.

Lemma 2.34.

Let $\mu_{0}$ be a probability measure on $\partial X$ that satisfies the shadow lemma as in Lemma 2.31, and let $A\subseteq\Gamma$ be a subset so that the partial Poincaré series $\mathcal{P}_{A}(\omega_{\Gamma},x,y)<\infty$ . Then the following set

\Lambda:=\limsup_{g\in A}\Pi_{o}^{F}(go,r)

is a $\mu_{o}$ –null set, for any $r\gg 0$ .

Proof.

By definition, we can write $\Lambda=\cap_{n\geq 1}\Lambda_{n}$ where

\Lambda_{n}=\bigcup_{g\in A:d(o,go)\geq n}\Pi_{o}^{F}(go,r)

If $\mu_{1}(\Lambda)>0$ then $\mu_{1}(\Lambda_{n})>\mu_{1}(\Lambda)/2$ for all $n\gg 0$ . On the other hand, the shadow lemma implies

\mu_{o}(\Lambda_{n})\leq\sum_{g\in A:d(o,go)\geq n}\mu_{o}(\Pi_{o}^{F}(go,r))\leq\sum_{g\in A:d(o,go)\geq n}\mathrm{e}^{-\omega_{\Gamma}d(o,go)}

so the tails of the series $\mathcal{P}_{A}(\omega_{\Gamma},x,y)$ has a uniform positive lower bound, contradicting to the assumption. The proof is complete. ∎

We list the following useful properties from [72, Lemma 4.6, Theorem 1.10, Lemma 5.5] about conical points.

Lemma 2.35.

The following holds for any $\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ :

(1)

$\xi$ is visual: for any basepoint $o\in\mathrm{X}$ there exists a geodesic ray starting at $o$ ending at $[\xi]$ .
(2)

If $G\curvearrowright\mathrm{X}$ is of divergence type, $\mu_{o}$ charges full measure on $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ and every $[\xi]$ –class is $\mu_{o}$ –null.
(3)

Let $y_{n},z_{n}\in\mathrm{X}$ be two sequences tending to $[\xi]$ in $\overline{\mathrm{X}}=\mathrm{X}\cup\partial{\mathrm{X}}$ . Then for any $x\in\mathrm{X}$ ,

$\limsup_{n\to\infty}|B_{y_{n}}(x)-B_{z_{n}}(x)|\leq 20C.$

In particular, $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ is a subset of $\mathcal{C}^{\mathrm{hor}}_{20C}$ (defined in Assumption D).

It is well known that in hyperbolic spaces, a horoball centered at a point in the Gromov boundary has the unique limit point at the center. This could be proved for uniquely ergodic points in Thurston boundary of Teichmüller spaces. Analogous to these facts, we have the following.

Lemma 2.36.

Let $\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ . Consider a horoball $\mathcal{HB}([\xi])$ centered at $[\xi]$ defined as Definition 2.22. Then any escaping sequence $x_{n}\in\mathcal{HB}([\xi])$ has the accumulation points in $[\xi]$ .

Proof.

According to definition, there exists a sequence of elements $g_{n}\in\Gamma$ so that $\xi\in\Pi_{r}^{F}(g_{n}o)$ . By [72, Lemma 4.4], there exists a geodesic ray $\gamma$ starting from $o$ terminating at $[\xi]$ so that

\|\gamma\cap N_{r}(g_{n}\mathrm{Ax}(f_{n}))\|\geq L

where $f_{n}\in F$ and $L:=\min\{d(o,fo):f\in F\}-2r$ .

Fix any $X:=g_{n}\mathrm{Ax}(f_{n})$ , and consider the exit point $y$ of $\gamma$ at $N_{C}(X)$ . Let $z_{m}\in\gamma\to[\xi]$ as $m\to\infty$ . Given $x_{n}\in\mathcal{HB}([\xi])$ , the following holds for $m\gg 0$ by Lemma 2.35(3):

(20)

B_{z_{m}}(x_{n})=d(x_{n},z_{m})-d(o,z_{m})\leq 21C.

We claim that the projection $y_{n}\in X$ of all but finitely many $x_{n}$ to $X$ lies in a $10C$ –neighborhood of $y$ . Indeed, if not, $d(y_{n},y)>10C$ holds for infinitely many $n$ . The contracting property of $X$ implies that $[z_{m},x_{n}]$ intersects $N_{C}(X)$ for all $n,m\gg 0$ , so $d(y,[x_{n},x_{m}])\leq 2C$ . Combined with (20), we obtain $d(y,x_{n})\leq d(y,o)+100C$ . This is a contradiction, as $x_{n}$ is a unbounded sequence in a fixed ball around $y$ .

If $L>100C$ is assumed, the above claim implies the corresponding projections of $o$ and $x_{n}$ to $X$ have distance at least $90C$ , so the contracting property shows $[o,x_{n}]\cap N_{C}(X)\neq\emptyset$ . That is, $x_{n}\in\Omega_{o}(N_{C}(X))$ for infinitely many $x_{n}$ . As $X=g_{n}\mathrm{Ax}(f_{n})$ is chosen arbitrarily along $\gamma$ , the assumption (B) in Definition 2.13 implies that $x_{n}$ and $g_{n}o$ have the same limit, so $x_{n}\to[\xi]$ . ∎

Uniqueness of quasi-conformal measures

Suppose that $\Gamma\curvearrowright\mathrm{X}$ is of divergence type, where $\mathrm{X}$ is compactified by the horofunction boundary $\partial_{h}{\mathrm{X}}$ . The quasi-conformal measures on $\partial_{h}{\mathrm{X}}$ are in general neither ergodic nor unique. To remedy this, we shall push the measures to the reduced horofunction boundary $[\partial_{h}{\mathrm{X}}]$ , modding out the finite difference relation. However, $[\partial_{h}{\mathrm{X}}]$ is a pathological topological object (i.e. may be non-Hausdorff). The remedy is to consider the reduced Myrberg limit set, which provides better topological property. Define

(21)

\Lambda^{\mathrm{Myr}}{(\Gamma o)}=\bigcap\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}

where the intersection is taken over all possible choice of three independent contracting elements $F$ in $\Gamma$ . By [72, Lemma 2.27 & Lemma 5.6], the quotient map

	$\displaystyle\Lambda^{\mathrm{Myr}}{(\Gamma o)}\quad\longrightarrow$	$\displaystyle\quad[\Lambda^{\mathrm{Myr}}{(\Gamma o)}]$
	$\displaystyle\xi\quad\longmapsto$	$\displaystyle\quad[\xi]$

is a closed map with compact fibers, thus $[\Lambda^{\mathrm{Myr}}{(\Gamma o)}]$ is a completely metrizable and second countable topological space.

Let $\{\mu_{x}\}_{x\in\mathrm{X}}$ be a $\omega_{\Gamma}$ –dimensional $\Gamma$ –equivariant conformal density on $\partial_{h}{\mathrm{X}}$ . This is pushed forward to a $\omega_{\Gamma}$ –dimensional $\Gamma$ –quasi-equivariant quasi-conformal density, denoted by $\{[\mu_{x}]\}_{x\in\mathrm{X}}$ , on $[\Lambda^{\mathrm{Myr}}{(\Gamma o)}]$ .

Lemma 2.37 ([72, Lemma 8.4]).

Let $\{[\mu_{x}]\}_{x\in\mathrm{X}}$ and $\{[\mu_{x}^{\prime}]\}_{x\in\mathrm{X}}$ be two $\omega_{\Gamma}$ –dimensional $\Gamma$ –quasi-equivariant quasi-conformal densities on the quotient $[\Lambda^{\mathrm{Myr}}{(\Gamma o)}]$ . Then the Radon-Nikodym derivative $d[\mu_{o}]/d[\mu_{o}^{\prime}]$ is bounded from above and below by a constant depending only on $\lambda$ in Definition 2.24.

2.8. Regularly contracting limit points

In this subsection, we first introduce a class of statistically convex-cocompact actions in [71] as a generalization of convex-cocompact actions, encompassing Examples in 1.2. This notion was independently introduced by Schapira-Tapie [63] as strongly positively recurrent manifold in a dynamical context.

Given constants $0\leq M_{1}\leq M_{2}$ , let $\mathcal{O}_{M_{1},M_{2}}$ be the set of elements $g\in\Gamma$ such that there exists some geodesic $\gamma$ between $N_{M_{2}}(o)$ and $N_{M_{2}}(go)$ with the property that the interior of $\gamma$ lies outside $N_{M_{1}}(\Gamma o)$ .

Definition 2.38 (SCC Action).

If there exist positive constants $M_{1},M_{2}>0$ such that $\omega_{\mathcal{O}_{M_{1},M_{2}}}<\omega_{\Gamma}<\infty$ , then the proper action of $\Gamma$ on $\mathrm{X}$ is called statistically convex-cocompact (SCC).

Remark 2.39.

The motivation for defining the set $\mathcal{O}_{M_{1},M_{2}}$ comes from the action of the fundamental group of a finite volume negatively curved Hadamard manifold on its universal cover. In that situation, it is rather easy to see that for appropriate constants $M_{1},M_{2}>0$ , the set $\mathcal{O}_{M_{1},M_{2}}$ coincides with the union of the orbits of cusp subgroups up to a finite Hausdorff distance. The assumption in SCC actions is called the parabolic gap condition by Dal’bo, Otal and Peigné in [25]. The growth rate $\omega_{\mathcal{O}_{M_{1},M_{2}}}$ is called complementary growth exponent in [5] and entropy at infinity in [63].

SCC actions have purely exponential growth, and thus are of divergence type. Therefore, we have the unique conformal density on a convergence boundary by Lemma 2.37.

Lemma 2.40.

Suppose that $\Gamma\curvearrowright\mathrm{X}$ is a non-elementary SCC action with contracting elements. Then $\Gamma$ has purely exponential growth: for any $n\geq 1$ , $\sharp N_{\Gamma}(o,n)\asymp\mathrm{e}^{\omega_{\Gamma}n}$ .

For the remainder of this subsection, assume that $\Gamma\curvearrowright\mathrm{X}$ is an SCC action with a contracting element. Fix a set $F$ of contracting elements in $\Gamma$ .

We now recall another specific class of conical limit points, introduced in [57], called regularly contracting limit points, which have been shown to be generic for PS measures there. This notion is modeled on the following purely metric notion of regularly contracting geodesics introduced earlier in [32]. For any ratio $\theta\in[0,1]$ , a $\theta$ –interval of a geodesic segment $\gamma$ means a connected subsegment of $\gamma$ with length $\theta\ell(\gamma)$ .

Definition 2.41.

Fix constants $\theta,r,C,L>0$ . We say that a geodesic $\gamma$ is $(r,C,L)$ –contracting at $\theta$ –frequency if every $\theta$ –interval of $\gamma$ contains a segment of length $L$ that is $r$ –close to a $C$ –contracting geodesic.

A geodesic ray $\gamma$ is $(r,C,L)$ –contracting at $\theta$ –frequency if any sufficiently long initial segment of $\gamma$ (i.e. $\gamma[0,t]$ for $t\gg 0$ ) is $(r,C,L)$ –contracting at $\theta$ –frequency.

Furthermore, $\gamma$ is frequently $(r,C,L)$ –contracting if it is $(r,C,L)$ –contracting at $\theta$ –frequency for any $\theta\in(0,1)$ .

The above notion is not used in this paper, but motivates the following analogous notion involving a proper action $\Gamma\curvearrowright\mathrm{X}$ .

Definition 2.42.

Fix $\theta\in(0,1]$ and $r>0$ and $f\in\Gamma$ . We say that a geodesic $\gamma$ contains $(r,f)$ –barriers at $\theta$ –frequency if for every $\theta$ –segment of $\gamma$ has $(r,f)$ –barriers.

An element $g\in G$ has $(r,f)$ –barriers at $\theta$ –frequency if there exists a geodesic $\gamma$ between $B(o,M)$ and $B(go,M)$ such that $\gamma$ has $(r,f)$ –barriers at $\theta$ –frequency.

Let $\mathcal{W}(\theta,r,F)$ denote the set of elements in $\Gamma$ having no $(r,f)$ –barriers at $\theta$ –frequency for some $f\in F$ . That is, an element $g$ of $\Gamma$ belongs to $\mathcal{W}(\theta,r,F)$ if and only if any geodesic between $B(o,M)$ and $B(go,M)$ contains a $\theta$ –interval that has no $(r,f)$ –barriers.

Lemma 2.43.

[57, Lemma 4.7] Fix $\theta\in(0,1],r>M$ . Then $\mathcal{W}(\theta,r,F)$ is growth tight.

Set $V:=\Gamma\setminus\mathcal{W}(\theta,r,F)$ . Let $\mathcal{FC}(\theta,r,F)$ denote the set of limit points $\xi\in\Lambda(\Gamma o)$ that is contained in infinitely many shadows at elements in $V$ . More precisely, $\mathcal{FC}({\theta,r,F})$ is the limit superior of the following sequence, as $n\to\infty$ ,

\bigcup\limits_{v\in A_{\Gamma}(o,n,\Delta)\cap{V}}\Pi_{o}^{F}(v,r)

where $A_{\Gamma}(o,n,\Delta)$ is defined in (16). Hence,

\Lambda_{n}:=\bigcup\limits_{k\geq n}\Bigg{(}\bigcup\limits_{v\in A_{\Gamma}(o,k,\Delta)\cap{V}}\Pi_{o}^{F}(v,r)\Bigg{)}\searrow\mathcal{FC}(\theta,r,F)\quad\textrm{as $n\rightarrow\infty$}.

Recall $F^{n}=\{f^{n}:f\in F\}$ for an integer $n\geq 1$ . Define the set of regularly contracting points as follows

\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}:=\bigcap_{n\in\mathbb{N}}\bigcap_{\theta\in(0,1]\cap\mathbb{Q}}\mathcal{FC}(\theta,r,F^{n})

We could take a further countable intersection over all possible $F$ of three independent contracting elements. In general, this is a proper subset of Mryberg set in [72].

It is proved in [57, Lemma 4.12] that the set of $(r,C,L)$ –regularly contracting rays lies in $\mathcal{FC}(\theta,r,F)$ for certain $F$ . The following result thus implies [57, Theorem A], saying that $(r,C,L)$ –regularly contracting rays is $\mu_{o}$ –full.

Proposition 2.44.

[57] Assume that $\Gamma\curvearrowright\mathrm{X}$ is an SCC action with contracting elements. Then the regularly contracting limit set $[\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}]$ is a $\mu_{o}$ –full subset of $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ .

Proof.

Recall that $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ is the limit super of $\{\Pi_{o}^{F}(go,r):g\in\Gamma\}$ , and it is $\mu_{o}$ –full by Lemma 2.35. The set $W=\mathcal{W}(\theta,r,F)$ is growth tight by Lemma 2.43, so the limit super $\Lambda$ of $\{\Pi_{o}^{F}(go,r):g\in W\}$ is $\mu_{o}$ –null by Lemma 2.34. Noting $V=\Gamma\setminus W$ as in the above definition, $[\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}]$ is contained in the complementary subset of $\Lambda$ in $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ . Hence, $\mu_{o}([\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}])=1$ is proved. ∎

Remark 2.45.

If $\mathrm{X}$ is a negatively curved Riemannian manifold with finite BMS measure on geodesic flow, then the PS measure is supported on the frequently contracting limit points. The same is true for certain CAT(0) groups with rank-1 elements and mapping class groups in [32, Theorems 5.1 & 7.1]. In these settings, finiteness of the BMS measure is equivalent to having purely exponential growth ([61]).

In the current coarse setting, we expect that the SCC action assumption could be replaced in Proposition 2.44 with purely exponential growth.

2.9. Hopf decomposition

In this subsection, we consider a (possibly non-discrete) countable subgroup $H<\mathrm{Isom}(\mathrm{X})$ . In particular, $H\curvearrowright\mathrm{X}$ is not necessarily a proper action with discrete orbits.

Assume $H$ admits a measure class preserving action on $(\partial{\mathrm{X}},m)$ . We say that the action $H\curvearrowright(\partial{\mathrm{X}},m)$ is (infinitely) conservative, if for any $A\subseteq\partial{\mathrm{X}}$ with $m(A)>0$ , there are (infinitely many) $h\in H\setminus\{1\}$ so that $hA\cap A\neq\emptyset.$ It is called dissipative if there is a measurable wandering set $A$ : $gA\cap A=\emptyset$ for each $1\neq h\in H$ . If in addition, $\cup_{h\in H}hA=\partial{\mathrm{X}}$ , then it is completely dissipative.

Let $\mathbf{C}\subseteq\partial{\mathrm{X}}$ be the union of purely atomic ergodic components, and $\mathbf{D}=\partial{\mathrm{X}}\setminus\mathbf{C}$ . Denote $\mathbf{D}_{1}$ (resp. $\mathbf{D}_{>1}$ ) the subset of $\mathbf{D}$ with trivial (resp. nontrivial) stabilizers, $\mathbf{D}_{\infty}$ the subset of $\mathbf{D}$ with infinite stabilizers. The partition $\mathbf{Cons}=\mathbf{C}\cup\mathbf{D}_{>1}$ and $\mathbf{Diss}=\mathbf{D}_{1}$ (mod 0) gives the Hopf decomposition $H\curvearrowright(\partial{\mathrm{X}},m)$ as the disjoint union of conservative and completely dissipative components. If $H\curvearrowright\mathrm{X}$ is proper on a hyperbolic space, $\mathcal{D}_{\infty}$ consists of parabolic points with infinite stabilizer. If $H$ is torsion-free, parabolic points are countable.

By [40, Theorem 29], the infinite conservative part $\mathbf{Cons}_{\infty}=\mathbf{C}\cup\mathbf{D}_{\infty}$ coincides (mod 0) with the following set

	$\displaystyle\mathbf{Cons}_{\infty}$	$\displaystyle=\{\xi\in\partial{\mathrm{X}}:\exists t>0\sharp\{dg_{\star}m(\xi)/dm(\xi)>t\}=\infty\}$
		$\displaystyle=\{\xi\in\partial{\mathrm{X}}:\sum_{g\in H}dg_{\star}m(\xi)/dm(\xi)=\infty\}$

We now consider the $\Gamma$ –equivariant $\omega_{\Gamma}$ –dimensional conformal density $\{\mu_{x}:x\in\mathrm{X}\}$ on $\partial{\mathrm{X}}$ . We emphasize that $H$ is not necessarily contained in $\Gamma$ , but is assumed to preserve the measure class of $m:=\mu_{o}$ , so that the following holds for $g_{\star}\mu_{o}=\mu_{go}$ :

\frac{dg_{\star}\mu_{o}}{d\mu_{o}}(\xi)\asymp\mathrm{e}^{-\omega_{\Gamma}B_{\xi}(go,o)}

Recall that $\mu_{o}$ is supported on $[\Lambda(\Gamma o)]$ .

We do not assume $H$ to be a discrete group, so $Ho$ may not be a discrete subset in $\mathrm{X}$ . We still denote by $\Lambda(Ho)$ the accumulation points of the orbit $Ho$ in the convergence boundary $\partial{\mathrm{X}}$ . It depends on the basepoint $o$ , but $[\Lambda(Ho)]$ does not, because of Assumption (B) in Definition 2.13.

Let $\mathcal{C}^{\mathrm{hor}}$ be defined in Assumption D as the set of points $\xi\in\mathcal{C}$ at which Busemann cocycles satisfy some weak form of continuity. According to the above discussion, $\mathbf{Cons}_{\infty}$ is exactly the so-called big horospheric limit point defined as follows.

Definition 2.46.

A point $\xi\in[\Lambda(Ho)]\cap\mathcal{C}^{\mathrm{hor}}$ is called big horospheric limit point if there exists $h_{n}\in H$ such that $B_{[\xi]}(h_{n}o,o)\geq L$ for some $L\in\mathbb{R}$ . If, in addition, $B_{[\xi]}(h_{n}o,o)\geq L$ holds for any given $L\in\mathbb{R}$ , then $\xi$ is called small horospheric limit point. Denote by $\Lambda^{\mathrm{Hor}}{(Ho)}$ (resp. $\Lambda^{\mathrm{hor}}{(Ho)}$ ) the set of the big (resp. small) horospheric limit points.

Remark 2.47.

By definition, a big or small horospheric limit point is a property of a $[\cdot]$ –class. Thus, $\Lambda^{\mathrm{Hor}}{(Ho)}=[\Lambda^{\mathrm{Hor}}{(Ho)}]$ is $[\cdot]$ –saturated. In terms of horoballs (2.22), $\xi$ is a big (resp. small) horospheric limit point if some (resp. any) horoball centered at $[\xi]$ contains infinitely many $h_{n}o$ with $h_{n}o\to\xi$ .

Let $\partial{\mathrm{X}}_{>1}$ (resp. $\partial{\mathrm{X}}_{1}$ ) denote the set of boundary points with nontrivial (resp. trivial) stabilizers in $H$ . Summarizing the above discussion, we have the Hopf decomposition for conformal measures.

Lemma 2.48.

Let $H$ act properly on $\mathrm{X}$ and preserve the measure class of $\mu_{o}$ . Then $\Lambda^{\mathrm{Hor}}{(Ho)}$ is the infinite conservative part, $\mathbf{Cons}=\Lambda^{\mathrm{Hor}}{(Ho)}\cup\partial{\mathrm{X}}_{>1}$ and $\mathbf{Diss}=\partial{\mathrm{X}}\setminus\mathbf{Cons}=\partial{\mathrm{X}}_{1}\setminus\Lambda^{\mathrm{Hor}}{(Ho)}$ .

It is wide open whether the big horospheric limit set differs from the small one only in a negligible set (cf. Question 1.18). This has been confirmed for Kleinian groups [65] for spherical measures, free subgroups [35] for visual measures and normal subgroups of divergent type actions [27] for general conformal measures without atoms. Here is an example of atomic conformal measures in which the large and small horospheric sets differ by a positive measure set.

Example 2.49.

Let $\Gamma$ be a non-uniform lattice in $\mathrm{Isom}(\mathbb{H}^{m})$ with $\omega_{\Gamma}=m-1\geq 1$ . We can put a conformal measure on the boundary at infinity, supported on the set $\mathbf{P}$ of countably many parabolic fixed points. Indeed, fix any $\omega>(m-1)$ and $\mu_{o}(\xi)=c>0$ for given $\xi\in\mathbf{P}$ . The value of $c$ will be adjusted below. The other points $\eta=g\xi$ in $\Gamma\xi$ is then determined by

\mu_{o}(\eta)=c\cdot\mathrm{e}^{-\omega B_{\xi}(g^{-1}o,o)}

where $\eta=g\xi$ which does not depend on $g\in\Gamma$ (for $B_{\xi}(\cdot,\cdot)$ is invariant under the stabilizer $\Gamma_{\xi}$ ). Fix $g_{\eta}\in\Gamma$ so that $g_{\eta}\xi=\eta$ . Observe that $\mu_{o}(\Gamma\xi)$ can realize any value in $(0,1]$ , by adjusting $c$ with the following fact for given $\omega>\omega_{\Gamma}$ ,

\sum_{\eta\in\Gamma\xi}\mathrm{e}^{-\omega B_{\eta}(o,g_{\eta}o)}<\infty

which, in turn, follows from purely exponential growth of double cosets of the parabolic subgroup $\Gamma_{\xi}$ (e.g. [36])

\sharp\{g\mathcal{HB}(\xi):d(\mathcal{HB}(\xi),g\mathcal{HB}(\xi))\leq n\}\asymp\mathrm{e}^{\omega_{\Gamma}n}

As $\sharp\mathbf{P}/\Gamma$ is finite, we are able to achieve that $\mu_{o}(\mathbf{P})=1$ .

The set of parabolic points is the infinite conservative part. If the number of cusps is at least 2, the action is not ergodic. Moreover, parabolic points are the big horospheric limit set (mod 0), but not small horospheric limit points. So $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=1$ and $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})=0$ .

We shall prove that $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)}\setminus\Lambda^{\mathrm{hor}}{(Ho)})=0$ for subgroups in a hyperbolic group, where $\mu_{o}$ is the Patterson-Sullivan measure for $\Gamma$ .

At last, we collect the notations of various limit sets studied in this paper:

•

$\Lambda(\Gamma o)$ is the limit set of $\Gamma o$ , while $[\Lambda(\Gamma o)]$ is the $[\cdot]$ –classes over $\Lambda(\Gamma o)$ .
•

$\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ is the set of $(r,F)$ –conical points, and $\Lambda^{\textrm{con}}{(\Gamma o)}$ denotes the set of usual conical points.
•

$\Lambda^{\mathrm{Hor}}(\Gamma o)$ (resp. $\Lambda^{\mathrm{hor}}(\Gamma o)$ ) is the big/small horospheric limit set.
•

$\Lambda^{\mathrm{Myr}}{(\Gamma o)}$ is the Myrberg limit set, and $\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}$ is the regularly contracting limit set.

They are related as

\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}\subseteq\Lambda^{\mathrm{Myr}}{(\Gamma o)}\subseteq\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}\subseteq\Lambda^{\textrm{con}}{(\Gamma o)}\subseteq\Lambda^{\mathrm{hor}}(\Gamma o)\subseteq\Lambda^{\mathrm{Hor}}(\Gamma o)\subseteq\Lambda(\Gamma o)

where each inclusion is proper in general.

Part I Hopf decomposition of confined subgroups

3. Prelude: proof of selected results in the hyperbolic setting

The goal of this section is two-fold: to illustrate some of our key tools via well-known facts in hyperbolic spaces; and present in this setting the proofs for (essentially all) theorems in §4 and §6 and a key tool used in subsequent sections. This section could be skipped without affecting the remaining ones.

Throughout this section, we assume that $\mathrm{X}$ is a Gromov hyperbolic space and $H$ , $\Gamma$ , $G$ are subgroups of $\mathrm{Isom}(\mathrm{X})$ with $H$ and $\Gamma$ being discrete. Let $\{\mu_{x}:x\in\mathrm{X}\}$ be Patterson-Sullivan measures on the Gromov boundary $\partial{\mathrm{X}}$ constructed from $\Gamma\curvearrowright\mathrm{X}$ . If $\Gamma\curvearrowright\mathrm{X}$ is of divergence type, then $\mu_{o}$ is atomless, and charges on the conical limit set (Lemma 2.35), and Shadow Lemma 2.31 holds. We assume the reader is familiar with these facts (e.g. which could be found in [50]).

Since $\mathrm{X}$ is Gromov hyperbolic, there exists $\delta>0$ such that any geodesic triangle admits a $\delta$ –center, that is, a point within a $\delta$ –distance to each side.

3.1. Preliminary

The notion of an admissible path (Def. 2.6) is a generalization of the $L$ –local quasi-geodesic paths:

p_{0}q_{1}p_{1}\cdots q_{n}p_{n}

where any two consecutive paths give a quasi-geodesic by (BP), of length at least $\ell(p_{i})>L$ (LL). It is well known that, in hyperbolic spaces, if $L\gg 0$ , a local quasi-geodesic is a global quasi-geodesic.

Assume that $\Gamma\curvearrowright\mathrm{X}$ is a proper non-elementary action, so there exist at least three loxodromic elements with pairwise disjoint fixed points. The following result, which is a special case of the Extension Lemma 2.10, is a consequence of the fact on quasi-geodesics mentioned above.

Lemma 3.1.

There exists a set $F$ of three loxodromic elements of $\Gamma$ and constants $L,c>0$ that depend only on $F$ with the following property. For any $g,h\in\Gamma$ , there exists an element $f\in F$ such that the path

\gamma:=[o,go]\cdot(g[o,fo])\cdot(gf[o,ho])

is a $c$ –quasigeodesic.

This result is well known to experts in the field and, to the best of our knowledge, appeared first in [6, Lemma 3]. It was subsequently reproved or implicitly used in many works, which we do not attempt to track down here.

3.2. Completely dissipative actions

In this subsection, we further assume that $\mathrm{X}$ is proper and $\Gamma$ acts on $\mathrm{X}$ cocompactly. The goal is to prove the conclusion of Theorem 4.2 in this setting.

Lemma 3.2.

Let $\xi\in\Lambda^{\mathrm{Hor}}{(Ho)}$ be a big horospheric point. Then for any $\epsilon>0$ , there exist infinitely many orbit points $h_{n}o\in Ho$ such that $d(o,z_{n})>d(o,h_{n}o)(1/2-\epsilon)$ , where $z_{n}$ is a $\delta$ –center of $\Delta(o,\xi,h_{n}o)$ for some $\delta>0$ .

Proof.

By the definition of a big horospheric point, there is an infinite sequence of points $h_{n}o\in Ho\to\xi$ such that $h_{n}o$ lies in some horoball $\mathcal{HB}(\xi,L)$ for all $n\geq 1$ and for some $L$ depending on $\xi$ . Recall the definition of a horoball: $\mathcal{HB}(\xi,L)=\{z\in\mathrm{X}:B_{\xi}(z,o)\leq L\}$ , so we obtain

B_{z}(h_{n}o,o)=d(z,h_{n}o)-d(z,o)\leq L+10\delta

for $z\in[o,\xi]$ with $d(o,z)$ sufficiently large. By hyperbolicity, if $z_{n}\in[o,\xi]$ is a $\delta$ –center of $\Delta(o,z,h_{n}o)$ for some $\delta>0$ , then $d(z,z_{n})+d(z_{n},h_{n}o)\leq 2\delta+d(z,h_{n}o)$ . Combining these inequalities gives

	$\displaystyle d(z,z_{n})+d(z_{n},h_{n}o)$	$\displaystyle\leq 12\delta+L+d(z,o)$
		$\displaystyle\leq 12\delta+L+d(o,z_{n})+d(z_{n},z)$

yielding $d(z_{n},h_{n}o)\leq 12\delta+L+d(o,z_{n})$ . Thus, $d(o,h_{n}o)\leq d(o,z_{n})+d(z_{n},h_{n}o)\leq 12\delta+L+2d(o,z_{n})$ . Note however that $L$ depends on $\xi$ . Given any $\epsilon>0$ , we can still obtain $(1/2-\epsilon)d(o,h_{n}o)\leq d(o,z_{n})$ after dropping finitely many $h_{n}o$ . The proof is complete. ∎

Theorem 3.3 (Theorem 1.10 in hyp. setup).

If $\omega_{H}<\omega_{\Gamma}/2$ , then $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=0$ .

Proof.

Fix a small $\epsilon>0$ , so $\omega_{H}<\omega_{\Gamma}(1/2-\epsilon)$ . Let $\hat{Z}$ be the set of points $z\in\mathrm{X}$ that satisfy $d(o,z)>d(o,ho)(1/2-\epsilon)$ , where $z$ is a $\delta$ center of the triangle $\Delta(o,ho,\xi)$ for some $h\in H$ and some $\xi\in\Lambda^{\mathrm{Hor}}{(Ho)}$ .

Choose a sufficiently large constant $r>18\delta$ such that the Shadow Lemma 2.31 applies. If $Z\subseteq\hat{Z}$ is a maximal $r$ –separated net in $\hat{Z}$ , then by Lemma 3.2, $\Lambda^{\mathrm{Hor}}{(Ho)}$ is contained in the limit superior $\limsup_{z\in Z}\Pi_{o}(z,2r)$ of shadows. As the action $\Gamma\curvearrowright\mathrm{X}$ is cocompact, we may assume $Z\subseteq\Gamma o$ upon increasing $r$ again.

Note that, for each $h\in H$ , the corresponding $z$ in the first paragraph is lying on the $\delta$ –neighborhood of $[o,ho]$ . This neighborhood contains at most $Cd(o,ho)$ points in the $r$ –separated set $Z$ where $C>0$ is a universal constant.

By the Shadow Lemma 2.31 for the $\Gamma$ –conformal density, we compute

	$\displaystyle\sum_{z\in Z}\mu_{o}(\Pi_{o}(z,2r))$	$\displaystyle\prec\sum_{h\in H}d(o,ho)\cdot\mathrm{e}^{-\omega_{\Gamma}d(o,ho)(1/2-\epsilon)}$
		$\displaystyle\prec\sum_{h\in H}\mathrm{e}^{-\omega d(o,ho)}<\infty$

which is finite by definition of $\omega_{H}$ and $\omega_{H}<\omega_{\Gamma}(1/2-\epsilon)$ . The Borel-Cantelli lemma implies that $\limsup_{z\in Z}\Pi_{o}(z,2r))$ is $\mu_{o}$ –null. Hence, $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=0$ , and the theorem is proved. ∎

3.3. Conservative actions for confined subgroups

This subsection is an abridged version of Sections §5 and §6 in hyperbolic setup. In this restricted setting, we will give two proofs that the horospheric limit set of a confined subgroup $H$ has full Patterson-Sullivan measure. One of the proofs will work whenever $H$ has a compact confining set, whereas the other requires a finite confining set. However, the latter proof is more readily adapted to the setting of general actions on metric spaces with contracting elements, which we will consider in Section 6. Therefore, we find it instructive to include both proofs.

We first state a key tool that allows us to use a finite confining set geometrically in hyperbolic spaces.

Lemma 3.4 ( $\ll$ Lemma 5.7).

Let $P\subseteq\mathrm{Isom}(\mathrm{X})$ be a finite set, so that no element in $P$ fixes pointwise the limit set $\Lambda(\Gamma o)$ . Then there exists a finite set $F\subseteq\Gamma$ of loxodromic elements and a constant $D>0$ with the following property.

For any $g,h\in G$ , one can find $f\in F$ so that for any $p\in P$ , the word $(g,f,p,f^{-1},h)$ labels a quasi-geodesic:

|d(o,gfpf^{-1}ho)-d(o,go)-d(o,ho)|\leq D

Proof.

Let $p\in P$ . By assumption, as all fixed points of loxodromic elements are dense in $\Lambda(\Gamma o)$ , $p$ moves the two fixed points of some $f$ . If $q\in P$ moves $g^{\pm}$ for another $g$ , then $p,q$ must move the two fixed points of $f^{n}g^{n}$ , which tends to $f^{+}$ and $g^{-}$ as $n\to\infty$ . As $P$ is a finite set, a common $f$ could be chosen for each $p\in P$ : $pf^{\pm}\cap f^{\pm}\neq\emptyset$ . Equivalently, $p\mathrm{Ax}(f)$ has bounded projection with $\mathrm{Ax}(f)$ for each $f\in P\setminus 1$ .

With a bit more effort, we produce $F=\{f_{1},f_{2},f_{3}\}$ where $p\in P$ moves the two endpoints of each $f\in F$ , and the set of axis $\{p\mathrm{Ax}(f):p\in P,f\in F\}$ has bounded projection. This implies that each $g\in\Gamma$ has bounded projection to at least two of $\mathrm{Ax}(f_{1}),\mathrm{Ax}(f_{2}),\mathrm{Ax}(f_{3})$ . Consequently, for any $g,h\in\Gamma$ , there exists a common $f\in F$ so that

\max\{\pi_{\mathrm{Ax}(f)}([o,go]),\pi_{\mathrm{Ax}(f)}([o,ho])\}\leq\tau

Set $L=\min\{d(o,fo):f\in F\}$ . The word $(g,f,p,f^{-1},h)$ labels an $L$ –local $c$ –quasi-geodesic path, denoted by $\gamma$ , where $c$ depends on $\tau$ . This concludes the proof. ∎

The following is an immediate consequence of Lemma 3.4 applied to $(g,g^{-1})$ , and $p\in P$ is chosen for $gf$ according to the definition of confined subgroups.

Lemma 3.5 ( $\ll$ Lemma 5.8).

Under the assumption of Lemma 3.4, for any $g\in G$ , there exist $f\in F$ and $p\in P$ such that $gfpf^{-1}g^{-1}$ lies in $H$ and

|d(o,gfpf^{-1}g^{-1}o)-2d(o,go)|\leq D

Recall that a point $\xi\in\Lambda^{\textrm{con}}{(\Gamma o)}$ is a conical point if there exist a sequence of $g_{n}\in\Gamma$ so that $g_{n}o$ lies in a $r$ –neighborhood of a geodesic ray $\gamma=[o,\xi]$ .

The next two results prove Theorem 1.11 in hyperbolic setup under the condition (i) and (ii) accordingly. The following does not use Lemmas 3.5 and 3.4.

Theorem 3.6 (Theorem 1.11(i) in hyp. setup).

Assume that $H<\mathrm{Isom}(\mathrm{X})$ is a torsion-free discrete subgroup confined by $\Gamma$ with a compact confining subset $P$ . Then the big horospheric limit set $\Lambda^{\mathrm{Hor}}{(Ho)}$ contains all but countably many points of $\Lambda^{\textrm{con}}{(\Gamma o)}$ .

Proof.

We denote by $\Lambda_{0}\in\Lambda^{\textrm{con}}{(\Gamma o)}$ the countable union of fixed points of all loxodromic and parabolic elements $h\in H$ . We shall prove that any conical point $\xi\in\Lambda^{\textrm{con}}{(\Gamma o)}\setminus\Lambda_{0}$ is contained in $\Lambda^{\mathrm{Hor}}{(Ho)}$ . By definition, there exists a sequence of $g_{n}\in\Gamma$ so that $g_{n}o$ lies in a $r$ –neighborhood of a geodesic ray $\gamma=[o,\xi]$ .

Set $D:=\max\{d(o,po):p\in P\}<\infty$ . The confined subgroup $H$ implies the existence of $p_{n}\in P$ so that $h_{n}:=g_{n}p_{n}g_{n}^{-1}\in H$ . Now, let $z\in[o,\xi]$ so that $d(g_{n}o,z)\leq r$ . Thus, $d(h_{n}o,z)\leq r+D+d(o,g_{n}o)\leq 2r+D+d(o,z)$ . Direct computation shows that $h_{n}o\in Ho$ lies in the horoball $\mathcal{HB}(\xi,o,2r+D+\epsilon)$ (or see Lemma 6.1 for this general fact).

A horoball $\mathcal{HB}(\xi)$ only accumulates at the center $\xi$ . As the orbit $Ho\subset X$ is discrete, it suffices to prove that $\{h_{n}o:n\geq 1\}$ is an infinite subset. Arguing by contradiction, assume now that $\{h_{n}o:n\geq 1\}$ is a finite set. By taking a subsequence, we may assume that $h:=h_{n}=h_{m}$ for any $n,m\geq 1$ .

By assumption, $H$ is torsion-free, so $p_{n}$ must be of infinite order. According to the classification of isometries, $p_{n}$ is either hyperbolic or parabolic, so is $h_{n}$ .

As $d(hg_{n}o,g_{n}o)=d(o,p_{n}o)<D$ , the convergence $g_{n}o\to\xi$ implies that $hg_{n}o\to\xi$ and then $h$ fixes $\xi$ . So $\xi\in\Lambda_{0}$ gives a contradiction, which completes the proof. ∎

We now demonstrate how to use Lemma 3.5 to deal with finite confining subset $P$ .

Theorem 3.7 (Theorem 1.11(ii) in hyp. setup).

Assume that $H<\mathrm{Isom}(\mathrm{X})$ is a discrete subgroup confined by $\Gamma$ with a finite confining subset $P$ . Assume that no element in $P$ fixes pointwise $\Lambda(\Gamma o)$ . Then the big horospheric limit set $\Lambda^{\mathrm{Hor}}{(Ho)}$ contains $\Lambda^{\textrm{con}}{(\Gamma o)}$ .

Proof.

Given $\xi\in\Lambda^{\textrm{con}}{(\Gamma o)}$ , there exists a sequence of $g_{n}\in\Gamma$ such that $d(g_{n}o,[o,\xi])\leq r$ . By Lemma 3.5, we can choose $f_{n}\in F$ and $p_{n}\in P$ so that $h_{n}:=g_{n}f_{n}p_{n}f_{n}^{-1}g_{n}^{-1}\in H$ labels a quasi-geodesic path.

If $d(g_{n}o,g_{m}o)\gg 0$ for any $n\neq m$ , then $h_{n}o\neq h_{m}o$ follows by Morse Lemma. Thus $\{h_{n}o:n\geq 1\}$ is an infinite subset. Setting

D=\max_{f\in F}\{d(o,fo)\}+\max_{p\in P}\{d(o,po)\}

we argue exactly as in the proof of Theorem 6.5 and obtain that $h_{n}o\in Ho$ lies in the horoball $\mathcal{HB}(\xi,2r+2D+\epsilon)$ . Hence, $\{h_{n}o:n\geq 1\}$ converges to $\xi$ , so $\xi$ is a big horospheric limit point. ∎

4. Completely dissipative actions

Our goal of this section is to prove Theorem 1.10 under the following setup.

•

The auxiliary proper action $\Gamma\curvearrowright\mathrm{X}$ is assumed to be SCC, in particular it is of divergence type.
•

Let $\partial{\mathrm{X}}$ be a convergence boundary for $\mathrm{X}$ and $\{\mu_{x}:x\in\mathrm{X}\}$ be the unique quasi-conformal, $\Gamma$ –equivariant density of dimension $\omega_{\Gamma}$ on $\partial{\mathrm{X}}$ .
•

Let $H<\mathrm{Isom}(\mathrm{X})$ be a discrete subgroup, which is confined by $\Gamma$ with a compact confining subset $P$ in $\mathrm{Isom}(\mathrm{X})$ .

We emphasize that $H$ is not necessarily contained in $\Gamma$ , but preserves the measure class of $\mu_{o}$ . This is motivated by the following example. If $\mathrm{X}$ is the rank-1 symmetric space equipped with the Lebesgue measure $\mu_{o}$ on the visual boundary, any subgroup $H<\mathrm{Isom}(\mathrm{X})$ preserves the measure class of $\mu_{0}$ . More generally, we have.

Lemma 4.1.

Suppose that $\mathrm{X}$ is a hyperbolic space on which $\Gamma<\mathrm{Isom}(\mathrm{X})$ acts properly and co-compactly. Let $\{\mu_{x}:x\in\mathrm{X}\}$ be the unique Patterson-Sullivan measure class of dimension $\omega_{\Gamma}$ on $\partial{\mathrm{X}}$ . Then $\mathrm{Isom}(\mathrm{X})$ preserves the measure class of $\mu_{o}$ .

Proof.

It is known that $\mu_{o}$ coincides with the Hausdorff measure of $\partial{\mathrm{X}}$ with respect to the visual metric. Namely, if $\rho_{\epsilon}$ is the visual metric for parameter $\epsilon$ , then $\mu_{o}(A)\asymp\mathcal{H}^{d}_{\rho_{\epsilon}}(A)$ where $d=\omega_{\Gamma}/\epsilon$ is the Hausdorff dimension. As an isometry on hyperbolic spaces induces a bi-Lipschitz map on boundary, it preserves the Hausdorff measure class. The proof is complete.∎

If $H$ contains no torsion, the set $\mathcal{D}_{<\infty}$ of points with nontrivial stabilizer is empty. Thus the conservative component is exactly the infinite conservative part, which by Lemma 2.48 coincides with the big horospherical limit set. Hence, we shall prove $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=0$ , provided that $\omega_{H}<\omega_{\Gamma}/2$ .

Our argument is essentially a geometric interpretation of the corresponding one in [35], where the same conclusion is proven for free groups. The proof in [35] is more combinatorial: it crucially uses the so-called Nielsen-Schreier system given by a spanning tree in the Schreier graph.

Recall that $\mathcal{C}_{\epsilon}^{\mathrm{hor}}$ is a subset in $\mathcal{C}$ (Assumption D) on which the Busemann cocycles converge up to an additive error $\epsilon$ in (8).

Theorem 4.2.

If $\omega_{H}<\omega_{\Gamma}/2$ , then $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=0$ . In particular, if $H$ is torsion-free, the action of $H$ on $(\partial{\mathrm{X}},\mu_{o})$ is completely dissipative.

Proof.

By Proposition 2.44, $\mu_{o}$ has full measure on $\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}$ , and $\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}$ is a subset of $\mathcal{C}^{\mathrm{hor}}_{20C}$ . By taking the intersection $\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}\cap\Lambda^{\mathrm{Hor}}{(Ho)}$ , we may assume in addition that every point $\xi\in\Lambda^{\mathrm{Hor}}{(Ho)}$ is a frequently contracting point. This is only the place in the proof where we use the SCC action.

For any sufficiently small $\theta\in(0,1]$ , instead of Lemma 3.2, we now prove

Claim.

Let $\xi\in\Lambda^{\mathrm{Hor}}{(Ho)}$ . There exist two sequences of elements $h_{n},t_{n}\in G$ such that

(22)		$\displaystyle d(o,t_{n}o)>(1/2-\theta)d(o,h_{n}o)$
(23)		$\displaystyle d(t_{n}o,[o,\xi])\leq r$
(24)		$\displaystyle d(t_{n}o,[o,h_{n}o])\leq C+\\|Fo\\|$

Proof of the claim.

As $\xi$ is a big horospheric point, there exists $h_{n}o\in\mathcal{HB}([\xi],L_{0})$ tending to $\xi$ , where $L_{0}$ depends on $\xi$ . That is, $B_{[\xi]}(h_{n}o,o)\leq L_{0}$ where $\xi\in\mathcal{C}^{\mathrm{hor}}_{20C}$ . Setting $M=L_{0}+20C$ , we obtain from (8):

(25)

\forall z\in[o,\xi]:\quad d(h_{n}o,z)\leq d(o,z)+M

Fix a big $L\gg M$ to be decided below, which also depends on $\xi$ .

Choose a point $z\in[o,\xi]$ so that $d(o,z)=d(o,h_{n}o)-L$ for $n\gg 0$ . Consider the initial segment $\alpha:=[o,z]$ of $[o,\xi]$ and one $\theta$ –interval $[x,y]\subseteq\alpha$ at the middle satisfying $d(o,x)=\ell(\alpha)/2$ and $d(y,\alpha_{+})=(1/2+\theta)\ell(\alpha)$ . By definition of $\xi\in\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}$ , any $\theta$ –interval of $\alpha$ contains $(r,f)$ –barrier $t\in G$ , so

\max\{d(to,[x,y]),d(tfo,[x,y])\}\leq r

where $f\in F$ is a contracting element. Up to enlarge $r$ , we assume $[to,tfo]\subseteq N_{r}([x,y]),$ so (23) is satisfied.

Figure 6. Schematic configuration in the proof of Theorem 4.2

Look at the triangle with vertices $(o,h_{n}o,z)$ for $z\in[o,\xi]$ . See Fig. (6). By the contracting property, $t\mathrm{Ax}(f)$ intersects non-trivially either $N_{C}([o,h_{n}o])$ or $N_{C}([z,h_{n}o])$ . Indeed, if this is false, the projection of $[o,h_{n}o]$ and $[z,h_{n}o]$ to $t\mathrm{Ax}(f)$ together has diameter at most $2C$ . Meanwhile, the corresponding initial and terminal segments of $\alpha$ entering and leaving $N_{C}(t\mathrm{Ax}(f))$ both project to $t\mathrm{Ax}(f)$ with diameter at most $C$ as well. Hence, we obtain $\|\alpha\cap N_{C}(t\mathrm{Ax}(f))\|\leq 4C$ . On the other hand, noting the above $\|\alpha\cap N_{r}(t\mathrm{Ax}(f))\|\geq d(o,fo)$ , we would get a contradiction if $d(o,fo)\gg 4C+2r$ .

Moreover, as $[to,tfo]\subseteq N_{r}(\alpha)\cap t\mathrm{Ax}(f)$ , the above argument further shows that $[to,tfo]$ intersects non-trivially either $N_{C}([o,h_{n}o])$ or $N_{C}([z,h_{n}o])$ .

We now claim that $[to,tfo]\cap N_{C}([z,h_{n}o])=\emptyset$ . Indeed, if not, we have $d(u,v)\leq C$ for some $u\in[to,tfo]$ and $v\in[z,h_{n}o]$ . Thus,

	$\displaystyle d(v,h_{n}o)$	$\displaystyle=d(z,h_{n}o)-d(z,v)$
		$\displaystyle\leq d(z,h_{n}o)-d(z,u)+C$
		$\displaystyle\leq d(z,o)-d(z,u)+C+M$

where the last inequality uses (25). On the other hand, as $d(o,h_{n}o)=d(o,z)+L$ , we have

	$\displaystyle d(v,h_{n}o)$	$\displaystyle\geq d(o,h_{n}o)-d(o,v)$
		$\displaystyle\geq d(o,z)-d(o,u)+L-C$

If $L>2C+M$ is assumed, this would give a contradiction, so the above claim is proved.

By the claim, let us choose $u\in[to,tfo]\cap N_{C}([o,h_{n}o])$ . Thus, we have $d(to,[o,h_{n}o])\leq C+d(o,fo)$ , so (24) is satisfied.

Recall that $\ell(\alpha)=d(o,h_{n}o)-L$ and $d(to,tfo)\leq d(x,y)+2r$ . For $u\in[to,tfo]$ , we have

d(u,to)\leq d(x,y)+2r\leq\theta\alpha+2r\leq\theta(d(o,h_{n}o)-L)+2r

and

d(o,u)\geq d(o,x)-r\geq\ell(\alpha)/2-r\geq(d(o,h_{n}o)-L)/2-r

which together give

d(o,to)\geq d(o,u)-d(u,to)\geq(d(o,h_{n}o)-L)(1/2-\theta)-3r.

As $d(o,h_{n}o)\to\infty$ , we may drop finitely many $h_{n}o$ to remove $L$ (which depends on $\xi$ though), so that $d(o,to)\geq d(o,h_{n}o)(1/2-2\theta).$ This shows that $t_{n}:=t$ is the desired element with (22,23,24). ∎

Now we choose $\theta,\epsilon>0$ so that $\omega_{H}+\epsilon<\omega_{\Gamma}(1/2-\theta)$ . Let $Z$ be the set of points $t\in\Gamma$ with the above property in the above Claim. Note that $\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}\cap\Lambda^{\mathrm{Hor}}{(Ho)}$ is contained in the limit superior of $\{\Pi_{o}^{F}(to,r):t\in Z\}$ . Note again that $h$ is over used by $t$ at most $C_{0}d(o,ho)$ times for a universal constant $C_{0}$ , as $to$ in (24) lies in a fixed neighborhood of $[o,ho]$ . Shadow lemma hence allows to compute

\sum_{t\in Z}\mu_{o}(\Pi_{o}^{F}(to,r))\leq\sum_{h\in H}d(o,ho)\mathrm{e}^{-\omega_{\Gamma}(1/2-\theta)d(o,ho)}\prec\sum_{h\in H}d(o,ho)\mathrm{e}^{-(\omega_{H}+\epsilon)d(o,ho)}<\infty.

Hence,

\mu_{o}(\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)}\cap\Lambda^{\mathrm{Hor}}{(Ho)})=0

by Borel-Cantelli Lemma. Thus, $\mu_{o}(\Lambda^{\textrm{reg}}_{\textrm{r,F}}{(\Gamma o)})=1$ implies $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=0$ , completing the proof. ∎

5. Preliminaries on confined subgroups

From this section to §9, we study the boundary actions of confined subgroups, growth and co-growth inequalities. In this section we first recall the notion of confined subgroups and review some known facts for later reference. Then we show an enhanced version of the Extension Lemma, which will be crucial in the later sections.

5.1. Confined subgroups

Definition 5.1.

Let $G$ be a locally compact, metrizable, topological group and $\Gamma<G$ . We say that $H<G$ is confined by $\Gamma$ if there exists a compact subset $P$ in $G$ such that $P$ intersects non-trivially with any conjugate $g^{-1}Hg$ for $g\in\Gamma$ :

g^{-1}Hg\cap(P\setminus\{1\})\neq\emptyset.

We refer $P$ as a confining subset for $H$ . If $\Gamma=G$ , we say that $H$ is a confined subgroup in $G$ .

In this paper, we are mainly interested in the case where $G=\mathrm{Isom}(\mathrm{X})$ or $G=\Gamma$ .

Note that any nontrivial normal subgroup is confined. Here are a few remarks in order.

Remark 5.2.

If $P$ is the confining subset, so is $P^{n}=\{p^{n}:p\in P\}$ for $n\geq 1$ . If $P$ is finite, we may further assume that $P$ is minimal: for any $p\in P$ , there exists $g\in G$ so that $gpg^{-1}\in H$ .

If $H$ is a confined subgroup in $G$ , then any subgroup containing $H$ is also confined. Note that being confined is inherited to finite index subgroups. Indeed, if $K$ is a finite index subgroup of $H$ , then any element $h\in H\setminus K$ admits a sufficiently large power in $K$ , so $K$ is a confined subgroup in $G$ .

Let $H$ be a discrete subgroup of $\mathrm{Isom}(\mathrm{X})$ . We define the injectivity radius of the action at a point $x\in\mathrm{X}$ as follows:

\textrm{Inj}_{H}(x)=\min\{d(x,hx):h\in H\setminus H_{x}\}

where $H_{x}:=\{h\in H:hx=x\}$ is finite by the proper action. By definition, $\textrm{Inj}_{H}(hx)=\textrm{Inj}_{H}(x)$ for $h\in H$ , so the injectivity radius function descends to the quotient $\mathrm{X}/H$ . When $H$ is torsion free, this coincides with the usual notion of injectivity radius, that is the maximal radius of the disk centered at $x$ in $\mathrm{X}$ injecting into $\mathrm{X}/H$ . It is clear that $\textrm{Inj}_{G}(Hx)\geq\textrm{Inj}_{G}(Gx)$ for any $H<G$ .

Lemma 5.3.

Assume that $\Gamma<\mathrm{Isom}(\mathrm{X})$ acts cocompactly (and not necessarily properly) on a $\Gamma$ –invariant subset $Z\subseteq\mathrm{X}$ . Denote $\pi:\mathrm{X}\to\mathrm{X}/H$ . Then the subset $\pi(Z)$ of $\mathrm{X}/H$ has injectivity radius bounded from above if and only if $H$ is confined by $\Gamma$ .

Proof.

(1). Suppose that $\textrm{Inj}_{H}:\mathrm{X}/H\to\mathbb{R}$ is bounded from above by $D>0$ over the set $\pi(Z)$ . That is, for any $x\in Z$ , there exists $h\in H\setminus H_{x}$ such that $d(x,hx)\leq D$ . Fixing a basepoint $o\in Z$ , as $\Gamma$ acts cocompactly on $X$ , there exists a constant $M>0$ such that $Z\subseteq N_{M}(\Gamma o)$ . Given $f\in\Gamma$ , let $x\in Z$ so that $d(fo,x)\leq M$ , so we have $d(o,f^{-1}hfo)\leq 2D+M$ . The set $P:=\{g\in\mathrm{Isom}(\mathrm{X}):d(o,go)\leq 2D+M\}$ is compact, so $H$ is confined by $\Gamma$ .

(2). Suppose that $H$ is confined by $\Gamma$ with a confining subset $P$ , we need to bound $\textrm{Inj}_{G}(Z/H)$ . For any $x\in Z$ , the cocompact action of $\Gamma$ on $Z$ implies the existence of $g\in\Gamma$ such that $d(go,x)\leq M$ . As $H$ is confined by $\Gamma$ , there exist $h\in H$ and $p\in P$ such that $g^{-1}hg=p$ . Thus, we have $d(x,hx)\leq d(go,hgo)+2M\leq d(o,po)+2M$ . Setting $D=2M+\max\{d(o,po):p\in P\}<\infty$ completes the other direction of the proof: $\textrm{Inj}_{H}(Hx)\leq D$ over $x\in Z$ . ∎

A direct corollary is the following.

Lemma 5.4.

Assume that a group $\Gamma<\mathrm{Isom}(\mathrm{X})$ acts cocompactly on $\mathrm{X}$ . Then for a discrete subgroup $H<G:=\mathrm{Isom}(\mathrm{X})$ , the following are equivalent:

(1)

$H$ is confined by $\Gamma$ ;
(2)

$H$ is confined in $G$ ;
(3)

$\mathrm{X}/H$ has injectivity radius bounded from above.

5.2. Elliptic radical

According to the definition, the product of any subgroup and a non-trivial finite normal subgroup is confined. To eliminate such pathological examples, we consider the notion of elliptic radical which, roughly speaking, consists of elements that fix the boundary pointwise.

Consider a convergence boundary $\partial{\mathrm{X}}$ of $\mathrm{X}$ , where the action of $\mathrm{Isom}(\mathrm{X})$ extends to $\partial{\mathrm{X}}$ by homeomorphisms and preserves the partition $[\cdot]$ (see Definition 2.13). Assume that $\Gamma$ is a non-elementary discrete subgroup of $\mathrm{Isom}(\mathrm{X})$ with non-pinched contracting elements.

Definition 5.5.

Let $G<\mathrm{Isom}(\mathrm{X})$ be a group. The elliptic radical $E_{G}(\Gamma)$ consists of elements $g\in G$ that induces an identity map on the limit set $[\Lambda(\Gamma o)]$ :

E_{G}(\Gamma):=\{g\in G:g[\xi]=[\xi],\forall[\xi]\subseteq[\Lambda(\Gamma o)]\}.

Note that $g\in E_{G}(\Gamma)$ may not fix pointwise $\Lambda(\Gamma o)$ . If $G=\mathrm{Isom}(\mathrm{X})$ we write $E(\Gamma)=E_{G}(\Gamma)$ .

Remark 5.6.

Recall that, given a contracting element $f\in\Gamma$ , $E_{G}(f)$ is the maximal elementary subgroup in $G$ that contains $f$ . By Lemma 2.17, if $f$ is non-pinched, then $E_{G}(f)$ is the set stabilizer of $[f^{\pm}]$ in $G$ .

If $G=\Gamma$ , $E_{G}(\Gamma)$ is the intersection of the maximal elementary subgroups $E_{G}(f)$ (in Lemma 2.2) for all contracting elements $f\in G$ . This is the unique maximal finite normal subgroup of $G$ .

5.3. An enhanced extension lemma

The following result will be a crucial tool in the next sections, which could be thought of as an enhanced version of Lemma 2.10.

Lemma 5.7.

Let $P\subseteq\mathrm{Isom}(\mathrm{X})$ be a finite set disjoint with $E(\Gamma)$ . Then there exists a finite set $F\subseteq\Gamma$ of contracting elements and constants $\tau,D>0$ with the following property. Set $L=\min\{d(o,fo):f\in F\}$ .

For any $g,h\in\Gamma$ , one can find $f\in F$ so that for any $p\in P$ , the word $(g,f,p,f^{-1},h)$ labels an $(L,\tau)$ –admissible path with associated contracting subsets $g\mathrm{Ax}(f)$ and $gfp\mathrm{Ax}(f)$ . Moreover,

|d(o,gfpf^{-1}ho)-d(o,go)-d(o,ho)|\leq D

Proof.

First of all, fixing a non-pinched contracting element $f\in\Gamma$ , we claim that for any $p\in P$ , there exists $g\in\Gamma$ so that $pg[f^{\pm}]\neq g[f^{\pm}]$ .

Indeed, if not, then $[pg\eta]=[g\eta]$ for any $g\in\Gamma$ and for any $\eta\in[f^{\pm}]$ . By Lemma 2.21, the set of $\Gamma$ –translates of its fixed points $[f^{\pm}]$ is dense in $\Lambda(\Gamma o)$ : $[\Lambda(\Gamma o)]=[\overline{\Gamma\eta}]$ . To be more precise, for any $\xi\in\Lambda(\Gamma o)$ , there exists $g_{n}\in\Gamma$ such that $g_{n}\eta\to\xi^{\prime}$ for some $\xi^{\prime}\in[\xi]$ . By assumption, $[pg_{n}\eta]=[g_{n}\eta]$ . According to Definition 2.13(C), $[\cdot]$ on $\Gamma[g^{\pm}]$ forms a closed relation, this implies $[p\xi^{\prime}]=[\xi^{\prime}]$ : that is, $p$ fixes every $[\cdot]$ –class in $[\Lambda(\Gamma o)]$ , so $p\in E(\Gamma)$ . This gives a contradiction. Hence, we see that $p\in P$ does not stabilize $g[f^{\pm}]$ for any $g\in\Gamma$ .

By Lemma 2.19, if $f,g$ are non-pinched contracting elements in $\Gamma$ , then $h_{n}:=f^{n}g^{n}$ for any $n\gg 0$ is a contracting element so that the attracting fixed point $[h^{+}]$ tends to $[f^{+}]$ and the repelling fixed point $[h^{+}]$ tends to $[g^{-}]$ as $n\to\infty$ . By the previous paragraphs, for any $p,q\notin E(\Gamma)$ , we can find two non-pinched contracting elements $f,g\in\Gamma$ , so that $p$ does not fix $[f^{+}]$ and $q$ does not fix $[g^{-}]$ . As $[f_{+}]$ and $[pf_{+}]$ are disjoint closed subsets in $\partial{\mathrm{X}}$ by Definition 2.13(A), choose disjoint open neighbourhoods such that Lemma 2.19 shows that $p,q$ do not stabilize the fixed points $[h_{n}^{\pm}]$ of $h_{n}$ for any $n\gg 0$ . We do not conclude here that $p,q\in E_{\Gamma}(h_{n})$ , as $E_{\Gamma}(h_{n})$ is the subgroup in $\Gamma$ (not in $\mathrm{Isom}(\mathrm{X})$ ) stabilizing $[h_{n}^{\pm}]$ .

As $P$ is a finite set, a repeated application of the previous paragraph produces an infinite set $\tilde{F}\subseteq\Gamma$ of independent contracting elements such that $p[f^{\pm}]\cap[f^{\pm}]=\emptyset$ for any $p\in P$ and $f\in\tilde{F}$ .

Now to conclude the proof, it suffices to invoke the same arguments in the proof of Lemma 2.10. Namely, for any finite set $F\subseteq\tilde{F}$ , there is a bounded intersection function $\tau:\mathbb{R}\to\mathbb{R}$ so that for any $f\neq f^{\prime}\in F$ and any $R>0$ ,

\|N_{R}(\mathrm{Ax}(f))\cap N_{R}(\mathrm{Ax}(f^{\prime}))\|\leq\tau(R).

The function $\tau$ depends only on the configuration of the axis $\mathrm{Ax}(f)=E_{\Gamma}(f)\cdot o$ for $f\in F$ . That is to say, $\tau$ remains the same if $F^{\prime}$ is chosen as a different set of representatives $f^{\prime}\in E_{\Gamma}(f)$ for $f\in F$ . Recall that $E_{\Gamma}(f)<\Gamma$ is the maximal elementary subgroup in $\Gamma$ defined in Lemma 2.2.

By the contracting property, for any $g\in\Gamma$ and any given $R\gg 0$ , the diameter $\|\pi_{\mathrm{Ax}(f)}([o,go])\|$ is comparable with $\|N_{R}(\mathrm{Ax}(f))\cap[o,go]\|$ . Hence, with at most one exception $f_{0}\in F$ ,

(26)

\forall f\in F\setminus\{f_{0}\}:\quad\pi_{\mathrm{Ax}(f)}([o,go])\leq\tau,

and, since $p\mathrm{Ax}(f)\neq\mathrm{Ax}(f)$ for any $f\in F$ and a finite set of $p\in P$ ,

(27)

\forall f\in F:\quad\pi_{\mathrm{Ax}(f)}([o,po])\leq\tau,

where $\tau=\tau(R)$ is a constant depending on the axis $\mathrm{Ax}(f)$ for $f\in F$ as above. Consequently, for any $g,h\in\Gamma$ , there exists a common $f\in F$ satisfying (26) so that

(28)

\max\{\pi_{\mathrm{Ax}(f)}([o,go]),\pi_{\mathrm{Ax}(f)}([o,ho])\}\leq\tau

Setting $L=\min\{d(o,fo):f\in F\}$ , the above equations (26)(27)(28) verify that for any $p\in P$ , the word $(g,f,p,f^{-1},h)$ labels an $(L,\tau)$ –admissible path, denoted by $\gamma$ , associated with the contracting sets $g\mathrm{Ax}(f)$ and $gfp\mathrm{Ax}(f)$ . Taking a high power of $f\in F$ if necessary, the constant $L$ can be large enough to satisfy Lemma 2.10, so any geodesic $\alpha$ with same endpoints as $\gamma$ $r$ –fellow travels $\gamma$ , so $go,gfo$ and $gfpo,gfpf^{-1}o$ have at most a distance $r$ to $\alpha$ . Letting $D=8r+L+\max\{d(o,po):p\in P\}$ concludes the proof of the lemma. ∎

5.4. First consequences on confined subgroups

Until the end of this subsection, consider a subgroup $H<G$ , which is confined by $\Gamma$ with a finite confining subset $P$ . Assume that $E_{G}(\Gamma)\cap P\setminus\{1\}\neq\emptyset$ . The main situations we keep in mind are $G=\mathrm{Isom}(\mathrm{X})$ or $G=\Gamma$ .

The following is an immediate consequence of Lemma 5.7 applied to $(g,g^{-1})$ , and $p\in P$ is chosen for $gf$ according to the definition of confined subgroups.

Lemma 5.8.

There exists a finite subset $F$ of contracting elements in $\Gamma$ with the following property: for any $g\in\Gamma$ , there exist $f\in F$ and $p\in P$ such that $gfpf^{-1}g^{-1}$ lies in $H$ and

|d(o,gfpf^{-1}g^{-1}o)-2d(o,go)|\leq D

where $D$ depends only on $F$ and $P$ .

This allows us to derive the following desirable property.

Lemma 5.9.

There exists a finite set $F$ of pairwise independent contracting elements in $\Gamma$ with the following property. Fix a conjugate $gHg^{-1}$ for $g\in\Gamma$ . For any $f_{1}\neq f_{2}\in F$ there exists $p_{1},p_{2}\in P$ such that $f_{i}p_{i}f_{i}^{-1}\in gHg^{-1}$ and their product $c(f_{1},f_{2}):=(f_{1}p_{1}f_{1}^{-1})\cdot(f_{2}p_{2}f_{2}^{-1})$ is a contracting element. Moreover, any two such $c(f_{1},f_{2})$ for distinct pairs $(f_{1},f_{2})\in F\times F$ are independent.

Proof.

For $i=1,2$ , applying Lemma 5.8 to $g^{-1}$ with $f_{i}$ yields the choice of $p_{i}$ in $P$ so that $h_{i}:=f_{i}p_{i}f_{i}^{-1}$ lies in $gHg^{-1}$ . It remains to note that $h:=h_{1}\cdot h_{2}$ is contracting. To this end, as $\mathrm{Ax}(f_{1})$ and $\mathrm{Ax}(f_{2})$ have bounded intersection, setting $L:=\min\{d(o,fo):f\in F\}$ , we have that the power $h^{n}$ labels an $(L,\tau)$ –admissible path as follows

\gamma:=\cup_{n\in\mathbb{Z}}h^{n}([o,f_{1}o]f_{1}[o,p_{1}o]f_{1}p_{1}[o,f_{1}^{-1}o]\cdot h_{1}[o,f_{2}o]f_{2}[o,p_{2}o]f_{2}p_{2}[o,f_{2}^{-1}o])

where the associated contracting subsets are given by the corresponding translated axis of $f_{1}$ and $f_{2}$ . Choosing $L$ sufficiently large, the path $\gamma$ is contracting, so $h$ is contracting.

Finally, if $h=c(f_{1},f_{2})$ and $h^{\prime}=c(f_{1}^{\prime},f_{2}^{\prime})$ for $(f_{1},f_{2})\neq(f_{1}^{\prime},f_{2}^{\prime})$ , then we need to show that $h$ and $h^{\prime}$ are independent. That amounts to proving that their axes $\gamma$ and $\gamma^{\prime}$ have infinite Hausdorff distance. Indeed, if not, then $\gamma$ lies in a finite neighborhood of $\gamma^{\prime}$ , so they fellow travel a common bi-infinite geodesic $\alpha$ (which could be obtained by applying Cantor argument with Ascoli-Arzela Lemma). For definiteness, say $f_{1}^{\prime}\neq f_{1}\neq f_{2}^{\prime}$ . The fellow travel property by Proposition 2.9 then forces a large intersection of some axis of $f_{1}$ with the axis of $f_{1}^{\prime}$ or the axis of $f_{2}^{\prime}$ . This would lead to a contraction with bounded intersection of $F$ . ∎

We get the following corollary on confined subgroups.

Lemma 5.10.

Assume that $H<G$ is a subgroup confined by $\Gamma$ with a finite confining subset $P$ that intersects trivially $E_{G}(\Gamma)$ . Then $H$ must be a non-elementary subgroup with a contracting element. Moreover, $\Lambda(\Gamma o)\subseteq[\Lambda(Ho)]$ .

Proof.

Let $g_{n}o\to\xi\in\Lambda(\Gamma o)$ for $g_{n}\in\Gamma$ . As above, we then choose $f_{n}\in F$ and $p_{n}\in P$ so that $h_{n}:=g_{n}f_{n}p_{n}f_{n}^{-1}g_{n}^{-1}\in H$ labels an $(L,\tau)$ –admissible path. Consider the sequence of contracting quasi-geodesic $X_{n}=g_{n}\mathrm{Ax}(f_{n})$ , so it follows that $h_{n}o\cap N_{r}(X_{n})\neq\emptyset$ where $r$ is given by Proposition 2.9. Hence, we proved that $g_{n}o,h_{n}o\in\Omega(N_{r}(X_{n}))$ , so the Assumption (B) in Definition 2.13 implies that $h_{n}o$ also tends to $[\xi]$ . The proof is now complete. ∎

6. Conservativity of boundary actions of confined subgroups

Our goal is to prove Theorem 1.11. In this section we retain the same setting as §4.

6.1. Some preliminary geometric lemmas

We first prepare some geometric lemmas. Recall that $\mathcal{C}_{\epsilon}^{\mathrm{hor}}$ is a subset in $\mathcal{C}$ (Assumption D) on which the Busemann cocycles converge up to an additive error $\epsilon$ in (8).

Lemma 6.1.

Let $\gamma=[o,\xi]$ be a geodesic ray ending at $[\xi]$ for some $\xi\in\mathcal{C}_{\epsilon}^{\mathrm{hor}}$ . If $y\in\mathrm{X}$ is a point that satisfies $d(y,z)\leq d(o,z)+D$ for some $z\in\gamma$ and a real number $D\in\mathbb{R}$ , then $y\in\mathcal{HB}(\xi,\epsilon+D)$ .

Proof.

According to Definition 2.22, the horoball $\mathcal{HB}(\xi,L)$ consists of points $x\in\mathrm{X}$ such that $B_{[\xi]}(x,o)\leq L$ . Noting that $d(\gamma(t),z)+d(z,o)=t$ for $z\in\gamma$ and $t\geq d(o,z)$ , so $d(\gamma(t),y)-t\leq[d(\gamma(t),z)+d(z,y)]-d(\gamma(t),z)-d(z,o)\leq d(z,y)-d(z,o)\leq D.$ Taking $t\to\infty$ shows $B_{[\xi]}(y,o)\leq\epsilon+D$ by (8). That is, $y\in\mathcal{HB}(\xi,\epsilon+D)$ . ∎

We now arrive at a crucial observation in the proof of Theorem 6.5.

Lemma 6.2.

Given $\tau,C>0$ there exists $L=L(C,\tau)$ with the following property. Assume that $\mathrm{Ax}(f)$ is $C$ –contracting. Consider a distinct axis $t\mathrm{Ax}(f)\neq\mathrm{Ax}(f)$ for some $t\in G$ . Let $g_{1}o\in\mathrm{Ax}(f),g_{2}o\in t\mathrm{Ax}(f)$ so that $[g_{1}o,g_{2}o]$ intersects $\mathrm{Ax}(f)$ in a diameter at least $L$ , and $g_{2}o$ lies in a $\tau$ –neighborhood of the projection of $\mathrm{Ax}(f)$ to $t\mathrm{Ax}(f)$ . Then $h:=g_{1}^{-1}g_{2}$ is a contracting element.

Proof.

The $C$ –contracting property of $\mathrm{Ax}(f)$ implies that any geodesic outside $N_{C}(\mathrm{Ax}(f))$ has $C$ –bounded projection, so $[o,ho]$ has $2C$ –bounded projection to $\mathrm{Ax}(f)$ . According to the assumption, $[o,ho]$ has $\tau$ –bounded projection to $h\mathrm{Ax}(f)$ , so this verifies that the path

\gamma=\cup_{n\in\mathbb{Z}}h^{n}[o,ho]

is an $(L,\tau+2C)$ –admissible path with associated contracting subsets $\{h^{n}\mathrm{Ax}(f),h^{n}t\mathrm{Ax}(f):n\in\mathbb{Z}\}$ . If $L$ is sufficiently large, then $\gamma$ is a contracting quasi-geodesic, concluding the proof that $g$ is contracting. ∎

As a corollary, we have.

Lemma 6.3.

Assume that $\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ is an $(r,F)$ –conical point. Let $[o,\xi]$ be a geodesic ray ending at $[\xi]$ with two distinct $(r,f)$ –barriers $X\neq Y$ . Let $g_{1}o\in X,g_{2}o\in Y$ be $C$ –close to the corresponding entry points of $[o,\xi]$ in $N_{C}(X)$ and $N_{C}(Y)$ . Then $g_{1}^{-1}g_{2}$ is a contracting element.

Recall that a geodesic metric space is geodesically complete, if any geodesic segment extends to a (possibly non-unique) bi-infinite geodesic. A smooth Hadamard manifold is geodesically complete.

Lemma 6.4.

Assume that $\mathrm{X}$ is a proper, geodesically complete, CAT(0) space. Let $p\in\mathrm{Isom}(\mathrm{X})$ be a parabolic isometry. If $p$ fixes a conical point $\xi$ in $\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ for a proper action of $\Gamma$ on $\mathrm{X}$ , then $p$ has the unique fixed point.

Proof.

By [30], under the assumption on $\mathrm{X}$ , the fixed point set $\mathrm{Fix}(p)$ of a parabolic element has diameter at most $\pi/2$ in the Tits metric on the visual boundary $\partial{\mathrm{X}}$ . A conical point $\xi$ is visible from any other point $\xi\neq\eta\in\partial{\mathrm{X}}$ ; that is, there is a bi-infinite geodesic between $\xi$ and $\eta$ . Thus, the angular metric between $\xi$ and $\eta$ is $\pi$ , so being the induced length metric, the Tits metric from $\xi$ to $\eta$ is at least $\pi$ . Hence, $\mathrm{Fix}(p)$ is singleton, completing the proof. ∎

6.2. Proof of Theorem 1.11

Assume that $H$ preserves the measure class of $\mu_{o}$ . By Lemma 2.35, $\mu_{o}$ is supported on the set of conical points $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ . By Lemma 2.35, $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ is a subset of $\mathcal{C}_{20C}^{\mathrm{hor}}$ (defined in Assumption D).

The conservativity of the action of $H$ on $(\partial{\mathrm{X}},\mu_{o})$ as stated in Theorem 1.11, follows from a combination of the next two theorems 6.5 and 6.7, which applies assuming the first condition (i) and second condition (ii) respectively. Note that there are situations (torsion-free $H$ with a finite confining set) where both theorems apply, but we stress that the arguments are of different flavors. The first result applies under the assumption of a compact confining subset $P$ without torsion; the second result crucially uses the Lemma 5.8 based on the finiteness of $P$ with torsion allowed.

Theorem 6.5.

Assume that $H<\mathrm{Isom}(\mathrm{X})$ is a torsion-free discrete confined subgroup. If $\mathrm{X}$ is neither hyperbolic nor geodesically complete CAT(0), assume, in addition, that $P$ is finite. Then the big horospheric limit set $\Lambda^{\mathrm{Hor}}{(Ho)}$ contains a $\mu_{o}$ –full subset of $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ .

Remark 6.6.

The proof shows that the conclusion of Theorem 6.5 remains valid for any atomless measure $\nu$ supported on $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ . If $\mathrm{X}$ is hyperbolic, $\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ could be replaced with $\Lambda^{\textrm{con}}{(\Gamma o)}$ . In particular, harmonic measures on horofunction boundary which can arise as the hitting measure of a random walk on $\Gamma$ are among such examples.

Proof.

Let $\Lambda_{0}$ be the countable union of fixed points $[h^{\pm}]$ of all contracting elements $h\in H$ . If $\mathrm{X}$ is hyperbolic or CAT(0), we adjoin into $\Lambda_{0}$ the fixed points of all parabolic elements in $H$ , which are countably many by Lemma 6.4. Note that $\mu_{o}(\Lambda_{0})=0$ for $\mu_{0}$ has no atom. By Lemma 2.35, we will prove the $\mu_{o}$ –full set of points $\xi\in[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]\setminus\Lambda_{0}$ is contained in $[\Lambda^{\mathrm{Hor}}{(Ho)}]$ . By definition of an $(r,F)$ –conical point, there exists a sequence of $(r,f_{n})$ –barriers $g_{n}\in\Gamma$ on a geodesic ray $\gamma=[o,\xi]$ , where $f_{n}\in F$ is taken over the finite set $F\subseteq\Gamma$ . By definition, we have $d(g_{n}o,[o,\xi]),d(g_{n}f_{n}o,[o,\xi])\leq r$ . Passing to a subsequence, we may assume that $f:=f_{n}$ for all $n\geq 1$ . Moreover, we can assume that $X_{n}$ are all distinct; otherwise $\xi$ lies in the limit set of the same $X_{n}$ for $n\gg 0$ , so is fixed by a contracting element $g_{n}fg_{n}^{-1}$ . This implies $\xi\in\Lambda_{0}$ , contradicting the assumption.

Let $P\subseteq\mathrm{Isom}(\mathrm{X},d)$ be a compact confining subset for $H$ , so $D:=\max\{d(o,po):p\in P\}<\infty$ .

Choose $p_{n}\in P$ so that $h_{n}:=g_{n}p_{n}g_{n}^{-1}\in H$ . Now, let $z\in[o,\xi]$ so that $d(g_{n}o,z)\leq r$ . Thus, $d(h_{n}o,z)\leq r+D+d(o,g_{n}o)\leq 2r+D+d(o,z)$ . According to Lemma 6.1, we see that $h_{n}o\in Ho$ lies in the horoball $\mathcal{HB}(\xi,o,2r+D+\epsilon)$ .

We shall prove that $\{h_{n}o:n\geq 1\}$ is an infinite (discrete) subset, which thus converges to $\xi$ by Lemma 2.36. Arguing by contradiction, assume now that $\{h_{n}o:n\geq 1\}$ is a finite set. By taking a subsequence, the proper action of $H$ allows us to assume that $h:=h_{n}=h_{m}$ for any $n,m\geq 1$ .

Case 1. $P$ is finite. We may then assume that $p_{n}=p$ for all $n\geq 1$ . We thus obtain $g_{m}^{-1}g_{n}h=hg_{m}^{-1}g_{n}$ . By Lemma 6.3, $g_{m}^{-1}g_{n}$ is a contracting element for any two $n\neq m\geq 1$ . If $h$ is of infinite order, $h\in E(g_{m}^{-1}g_{n})$ gives a contradiction, as any infinite order element in $E(g_{m}^{-1}g_{n})$ is a contracting element.

Case 2. $P$ may be an infinite compact set, but $\mathrm{X}$ is assumed to be hyperbolic or geodesically complete CAT(0). Note that $p_{n}$ might be distinct in general. As $H$ is torsion-free, $p_{n}$ must be of infinite order. According to classification of isometries, $p_{n}$ is either hyperbolic or parabolic.

As $d(hg_{n}o,g_{n}o)=d(o,p_{n}o)<D$ , the convergence $g_{n}o\to[\xi]$ implies that $hg_{n}o\to[\xi]$ and then $h$ fixes $[\xi]$ . So $\xi\in\Lambda_{0}$ gives a contradiction. The proof is complete. ∎

In the following statement, we emphasize that $H<\mathrm{Isom}(\mathrm{X})$ is not necessarily a discrete subgroup. The proof relies on Lemma 5.8 applied with $G=\mathrm{Isom}(\mathrm{X})$ here, which does not use the proper action $H\curvearrowright\mathrm{X}$ as well. Compared with Theorem 6.5, $H$ may contain torsion elements, but $P\cap E(\Gamma)$ is assumed to be trivial.

Theorem 6.7.

Assume that $H<\mathrm{Isom}(\mathrm{X})$ is a subgroup confined by $\Gamma$ with a finite confining subset $P$ that intersects trivially $E(\Gamma)$ . Then $\Lambda^{\mathrm{Hor}}{(Ho)}$ contains $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ as a subset. Moreover, $\Lambda(\Gamma o)\subseteq[\Lambda(Ho)]$ .

Proof.

By definition of an $(r,F)$ –conical point, there exists a sequence of $(r,F)$ –barriers $g_{n}\in\Gamma$ on a geodesic ray $\gamma=[o,\xi]$ . By definition, we have $d(g_{n}o,[o,\xi])\leq r$ .

By Lemma 5.8, we can choose $f_{n}\in F$ and $p_{n}\in P$ so that $h_{n}:=g_{n}f_{n}p_{n}f_{n}^{-1}g_{n}^{-1}\in H$ labels an $(L,\tau)$ –admissible path. If $d(g_{n}o,g_{m}o)\gg 0$ for any $n\neq m$ , then $h_{n}o\neq h_{m}o$ by Lemma 2.12. Thus $\{h_{n}o:n\geq 1\}$ is an infinite subset. Setting

D=\max_{f\in F}\{d(o,fo)\}+\max_{p\in P}\{d(o,po)\}

we argue exactly as in the proof of Theorem 6.5 and obtain that $h_{n}o\in Ho$ lies in the horoball $\mathcal{HB}([\xi],2r+2D+\epsilon)$ (the main issue there was proving the infiniteness of $\{h_{n}o:n\geq 1\}$ ). Hence, $\{h_{n}o:n\geq 1\}$ converges to $[\xi]$ by Lemma 2.36, so $\xi$ is a big horospheric limit point. ∎

Corollary 6.8.

In the setting of Theorems 6.5 or 6.7, if $H$ is a subgroup of $\Gamma$ , then $[\Lambda(\Gamma o)]=[\Lambda(Ho)]$ .

Proof.

Under the assumptions of Theorem 6.5, we proved that a $\mu_{o}$ –full subset $\Lambda$ of $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ is contained in the horospheric limit set $\Lambda^{\mathrm{Hor}}{(Ho)}$ . Taking a countable intersection $\Gamma:=\cap_{g\in\Gamma}g\Lambda$ allows to assume that $\Lambda$ is $\Gamma$ –invariant. If $\mathrm{X}$ is hyperbolic or CAT(0), it is well-known that the limit set $\Lambda(\Gamma o)$ is a $\Gamma$ –invariant minimal subset. Hence, the topological closure of $\Lambda$ recovers $\Lambda(\Gamma o)$ , thus verifying $\Lambda(\Gamma o)\subseteq\Lambda(Ho)$ , so the proof is completed in this case.

Otherwise, $P$ is a finite set by assumption. As $H<\Gamma$ is assumed, the conclusion follows immediately from Lemma 5.10. ∎

To conclude this subsection, we record a stronger statement, provided that $H$ is a normal subgroup. The proof strategy is due to [27].

Theorem 6.9.

Suppose that $H$ is an infinite normal subgroup of $\Gamma$ . Then $\Lambda^{\mathrm{hor}}{(Ho)}$ contains $\Lambda^{\mathrm{Myr}}{(\Gamma o)}$ as a subset.

Proof.

Given $\xi\in\Lambda^{\mathrm{Myr}}{(\Gamma o)}$ , let $\gamma$ be a geodesic ray starting at $o$ and ending at $[\xi]$ . Let us take a contracting element $f\in H$ , as an infinite normal subgroup contains infinitely many ones. By definition of $\Lambda^{\mathrm{Myr}}{(\Gamma o)}$ in (21), $\gamma$ contains infinitely many $(r,f^{n})$ –barriers $g_{n}$ for any $n\geq 1$ . By normality of $H$ , we have $g_{n}f^{n}g_{n}^{-1}\in H$ forms a sequence of elements, which enters any given horoball based on $[\xi]$ . With Lemma 2.36, this implies that $\Lambda^{\mathrm{Myr}}{(\Gamma o)}\subseteq[\Lambda^{\mathrm{hor}}{(Ho)}]$ . ∎

Part II Growth inequalities for confined subgroups

7. Shadow Principle for confined subgroups

We shall establish in this section a Shadow Principle for confined subgroups, which is crucial for the next two sections §8 and 9. The basic setup is as follows.

•

The auxiliary proper action $\Gamma\curvearrowright\mathrm{X}$ is assumed to be of divergence type with contracting elements.
•

Let $\partial{\mathrm{X}}$ be a convergence boundary for $\mathrm{X}$ and $\{\mu_{x}:x\in\mathrm{X}\}$ be a quasi-conformal, $\Gamma$ –equivariant density of dimension $\omega_{\Gamma}$ on $\partial{\mathrm{X}}$ .
•

Let $H$ be a discrete subgroup of a group $G<\mathrm{Isom}(\mathrm{X})$ , confined by $\Gamma$ , with a finite confining subset $P$ that intersects trivially $E(\Gamma)\setminus\{1\}$ .

Recall that $E(\Gamma)$ is the $[\cdot]$ –class stabilizer of $[\Lambda(\Gamma o)]$ in $\mathrm{Isom}(\mathrm{X})$ (Def. 5.5).

In this section, let $F\subseteq\Gamma$ be a set of three independent contracting elements, and the constants $L,\tau,r$ given by Lemma 5.7.

7.1. Shadow Principle for conformal density supported on boundary

Below, the constant $\lambda$ is given in Definition 2.24, and $\epsilon$ is the convergence error of the horofunction in Assumption D. Note that $\epsilon=0$ for the horofunction boundary. By Corollary 6.8, we have $[\Lambda(Ho)=[\Lambda(\Gamma o)]$ .

Lemma 7.1 (Shadow Principle).

Let $\{\mu_{x}\}_{x\in\mathrm{X}}$ be a $\omega_{H}$ –dimensional $H$ –quasi-equivariant quasi-conformal density supported on $\partial{\mathrm{X}}$ . Then there exists $r_{0}>0$ such that

\begin{array}[]{rl}\|\mu_{y}\|e^{-\omega_{H}\cdot d(x,y)}\quad\prec_{\lambda}\quad\mu_{x}(\Pi_{x}^{F}(y,r))\quad\prec_{\lambda,\epsilon,r}\quad\|\mu_{y}\|e^{-\omega_{H}\cdot d(x,y)}\\ \end{array}

for any $x,y\in\Gamma o$ and $r\geq r_{0}$ .

Proof.

We start by proving the lower bound, which is the key part of the proof. Write explicitly $x=g_{1}o$ and $y=g_{2}o$ for $g_{1},g_{2}\in\Gamma$ . Let $g=g_{1}^{-1}g_{2}$ .

1. Lower Bound. Fix a Borel subset $U:=[\Lambda(Ho)]$ such that $\mu_{o}(U)>0$ . For given $\xi\in U$ , there exists a sequence of points $h_{n}o\in\mathrm{X}$ such that $h_{n}o\to\xi$ . Since $F$ is a finite set, up to taking a subsequence of $h_{n}o$ , there exist $f\in F$ and $r>0$ given by Lemma 5.7 such that for any $p\in P$ , $g$ is an $(r,f)$ –barrier for any geodesic $[o,gfpf^{-1}h_{n}o]$ . This implies $gfpf^{-1}h_{n}o\in\Omega_{o}^{F}(go,r)$ , which tends to $gfpf^{-1}\xi$ . In addition, we can choose $p_{f}\in P$ according to definition of the confined subgroup $H$ so that $g_{2}fp_{f}(g_{2}f)^{-1}\in H$ . By definition of the shadow,

(29)

gfp_{f}f^{-1}\xi\in\Pi_{o}^{F}(go,r)

Note that $f\in F$ depends on $\xi\in U$ , and $F$ consists of three elements.

Consequently, the set $U$ can be decomposed as a disjoint union of three sets $U_{1},U_{2},U_{3}$ : for each $U_{i}$ , there exist $f_{i}\in F,p_{i}\in P$ such that $gf_{i}p_{i}f_{i}^{-1}U_{i}\subseteq\Pi_{o}^{F}(go,r)$ ; equivalently,

(30)

g_{2}f_{i}p_{i}f_{i}^{-1}U_{i}\subseteq\Pi_{g_{1}o}^{F}(g_{2}o,r).

Denote $L=\max\{2d(o,fo)+d(o,po):f\in F,p\in P\}<\infty$ . As $d(o,f_{i}p_{i}f_{i}^{-1}o)\leq L,$ there exists $\theta=\theta(L,\omega_{H})>0$ by quasi-conformality (12) such that

(31)

\mu_{g_{2}o}(g_{2}f_{i}p_{i}f_{i}^{-1}U_{i})\geq\theta\cdot\mu_{g_{2}f_{i}p_{i}f_{i}^{-1}o}(g_{2}f_{i}p_{i}f_{i}^{-1}U_{i}).

Figure 7. Upper bound in Shadow Principle: transporting a large set

U

into the shadow by Lemma 5.7.

We apply the $H$ –quasi-equivariance (11) with $g_{2}f_{i}p_{i}^{-1}f_{i}^{-1}g_{2}^{-1}\in H$ to the right-hand side:

(32)

\mu_{g_{2}f_{i}p_{i}f_{i}^{-1}o}(g_{2}f_{i}p_{i}f_{i}^{-1}U_{i})\geq\lambda^{-1}\cdot\mu_{g_{2}o}(g_{2}U_{i}).

Combining together (30), (31) and (32), we have

	$\displaystyle\mu_{g_{2}o}(\Pi_{g_{1}o}^{F}(g_{2}o,r))$	$\displaystyle\geq\sum_{1\leq i\leq 3}\mu_{g_{2}o}(g_{2}U_{i})/3$
		$\displaystyle\geq\lambda^{-1}\theta\cdot\mu_{g_{2}o}(U)/3$

where the last line uses the $H$ –invariance of $U=[\Lambda(Ho)]=[\Lambda(\Gamma o)]$ by Lemma 5.10.

Denote $M:=\lambda^{-2}\cdot\theta/3$ . We now conclude the proof for the lower bound via quasi-conformality (12):

\begin{array}[]{rl}\mu_{g_{1}o}(\Pi_{g_{1}o}^{F}(g_{2}o,r))&\geq\lambda^{-1}\cdot e^{-\omega\cdot d(g_{1}o,g_{2}o)}\cdot\mu_{g_{2}o}(\Pi_{g_{1}o}^{F}(g_{2}o,r))\\ \\ &\geq M\cdot\mu_{g_{2}o}(U)\cdot e^{-\omega\cdot d(g_{1}o,g_{2}o)}\end{array}

2. Upper Bound. Fix $r\geq r_{0}$ . Given $\xi\in\Pi_{g_{1}o}^{F}(g_{2}o,r)$ , there is a sequence of $z_{n}\in\mathrm{X}$ tending to $\xi$ such that $\gamma_{n}\cap B(g_{2}o,r)\neq\emptyset$ for $\gamma_{n}:=[g_{1}^{-1}o,z_{n}]$ . Since Buseman cocyles extend continuously to the horofunction boundary, we obtain

|B_{\xi}(g_{1}o,g_{2}o)-d(g_{1}o,g_{2}o))|\leq 2r.

The upper bound is given as follows:

\begin{array}[]{rl}\mu_{g_{1}o}(\Pi_{g_{1}o}^{F}(g_{2}o,r))&\leq\displaystyle\int_{\Pi_{g_{1}o}^{F}(g_{2}o,r)}e^{-\omega B_{\xi}(g_{1}o,g_{2}o)}d\mu_{g_{2}o}(\xi)\\ \\ &\leq\lambda e^{2\omega r}\|\mu_{g_{2}o}\|\cdot e^{-\omega d(g_{1}o,g_{2}o)}\end{array}

The proof of lemma is complete. ∎

7.2. Shadow Principle for conformal density supported inside

Fix a basepoint $o\in\mathrm{X}$ and $\omega>\omega_{H}$ . Given $x\in\mathrm{X}$ , define $\mu_{x}=\frac{1}{\mathcal{P}_{H}(\omega,o,o)}\sum_{h\in H}\mathrm{e}^{-\omega d(x,ho)}$ supported on $Ho$ , we have

\forall z\in Ho:\quad\frac{d\mu_{x}}{d\mu_{y}}(z)=\mathrm{e}^{-\omega[d(x,z)-d(y,z)]}

Lemma 7.2.

There are constants $\theta,D>0$ such that, given any $\Delta>0$ and, $g_{1},g_{2}\in G$ we have

\#H\cap A_{\Gamma}(g_{2}o,n,\Delta)\leq\theta\#H\cap A_{\Gamma}(g_{2}o,n,\Delta+D)\cap\Omega_{g_{1}o}^{F}(g_{2}o,D).

Proof.

Let $h\in H$ and $g:=g_{1}^{-1}g_{2}$ . According to Lemma 5.7, for given $g$ and $g_{2}^{-1}h$ , we pick $f_{h}\in F$ such that for any $p\in P$ , the word $(g,f_{h},p,f_{h}^{-1},g_{2}^{-1}h)$ labels an $(L,\tau)$ –admissible path, where $go$ is an $(r,F)$ –barrier for $f_{h}\in F$ . This implies $gf_{h}pf_{h}^{-1}g_{2}^{-1}h\in\Omega_{o}^{F}(go,r)$ , or equivalently $\mathbf{h}:=g_{2}f_{h}pf_{h}^{-1}g_{2}^{-1}h\in\Omega_{g_{1}o}^{F}(g_{2}o,r)$ . As $p\in P$ can be arbitrarily chosen, the definition of confined subgroup $H$ allows us to choose $p_{h}\in P$ for the element $g_{2}^{-1}h\in G$ so that $h^{-1}g_{2}f_{h}\cdot p_{h}\cdot(f_{h}^{-1}g_{2}^{-1}h)$ lies in $H$ and so $\mathbf{h}=g_{2}f_{h}p_{h}f_{h}^{-1}g_{2}^{-1}h$ lies in $H$ .

Set $D=\max\{d(o,fo)+d(o,po):f\in F,p\in P\}$ . If $d(ho,g_{2}o)\in[n-\Delta,n+\Delta]$ we have $d(\mathbf{h}o,g_{2}o)\in[n-\Delta-D,n+\Delta+D]$ . We claim that this assignment

	$\displaystyle\pi:H\cap A_{\Gamma}(g_{2}o,n,\Delta)$	$\displaystyle\longrightarrow H\cap A_{\Gamma}(g_{2}o,n,\Delta+D)\cap\Omega_{g_{1}o}^{F}(g_{2}o,r)$
	$\displaystyle h$	$\displaystyle\longmapsto\mathbf{h}$

is uniformly finite to one. Indeed, assume $\mathbf{h}=\mathbf{h^{\prime}}$ for some $h\neq h^{\prime}$ . Applying multiplication by $g_{1}^{-1}$ on the left for $\mathbf{h}=\mathbf{h^{\prime}}$ gives $gf_{h}p_{h}f_{h}^{-1}g_{2}^{-1}h=gf_{h^{\prime}}p_{h^{\prime}}f_{h^{\prime}}^{-1}g_{2}^{-1}h^{\prime}$ . As mentioned above, these two words label two $(L,\tau)$ –admissible paths with the same endpoints. Noting that $g^{-1}h,g^{-1}h^{\prime}\in A_{\Gamma}(o,n,\Delta)$ , Lemma 2.12 implies $d(g^{-1}ho,g^{-1}h^{\prime}o)=d(ho,h^{\prime}o)\leq R$ . Consequently, at most $N:=\sharp\{h\in H:d(o,ho)\leq R\}$ elements have the same image under the map $\pi$ . Setting $\theta=1/N$ thus completes the proof. ∎

The following is the Shadow Principle we want for all $s>w_{H}$ .

Lemma 7.3.

Fix any $\omega>\omega_{H}$ . Let $\{\mu_{x}\}_{x\in\mathrm{X}}$ be a $\omega$ –dimensional $H$ –quasi-equivariant quasi-conformal density supported on $Ho\subseteq\mathrm{X}$ . Then there exists $r_{0}>0$ such that

\begin{array}[]{rl}\|\mu_{y}\|e^{-\omega\cdot d(x,y)}\quad\prec_{\lambda}\quad\mu_{x}(\Omega_{x}^{F}(y,r))\quad\prec_{\lambda,\epsilon,r}\quad\|\mu_{y}\|e^{-\omega\cdot d(x,y)}\\ \end{array}

for any $x,y\in Go$ and $r\geq r_{0}$ .

Proof.

If $z\in\Omega_{x}^{F}(y,r)$ we have $d(x,z)+2r\geq d(x,y)+d(y,z)$ , so $\mu_{x}(\Omega_{x}^{F}(y,r))\leq\mathrm{e}^{2r\omega}\|\mu_{y}\|e^{-\omega\cdot d(x,y)}$ . Thus, it remains to prove the lower bound. As the triangle inequality implies $\mu_{x}(\Omega_{x}^{F}(y,r))\geq e^{-\omega d(x,y)}\mu_{y}(\Omega_{x}^{F}(y,r))$ , it suffices to show $\mu_{y}(\Pi_{x}^{F}(y,r))\geq\Theta\|\mu_{y}\|$ for some $\Theta>0$ .

By definition,

\mathcal{P}_{H}(\omega,o,o)\|\mu_{y}\|=\sum_{h\in H}\mathrm{e}^{-\omega d(y,ho)}<\infty

Assume that $x=g_{1}o,y=g_{2}o$ , and set $g=g_{1}^{-1}g_{2}$ for simplicity. Note that $\Omega_{x}^{F}(y,r)=g_{1}\Omega_{o}^{F}(go,r)$ . We remark, however, that $\mu_{y}(\Omega_{x}^{F}(y,r))=\mu_{go}(\Omega_{o}^{F}(go,r))$ may not hold, as $\mu_{x}$ is only $H$ –conformal density but $g_{1}$ may not be in $H$ .

By Lemma 7.2, we have

	$\displaystyle\sum_{h\in H}e^{-sd(g_{2}o,ho)}$	$\displaystyle\leq C_{1}(s)\sum_{n=0}^{\infty}e^{-sn}\#H\cap A(g_{2}o,n,\Delta)$
		$\displaystyle\leq\theta C_{1}(s)\sum_{n=0}^{\infty}e^{-sn}\cdot\#\left(H\cap A(g_{2}o,n,\Delta+D)\cap\Omega_{g_{1}o}^{F}(g_{2}o,r)\right)$
		$\displaystyle\leq\theta C_{1}(s)C_{2}(s)\sum_{h\in H\cap\Omega_{g_{1}o}^{F}(g_{2}o,r)}e^{-sd(g_{2}o,ho)}.$

Here, $C_{1}(s)=\Delta\cdot e^{-s\Delta},C_{2}(s)=(\Delta+D)\cdot e^{-s(\Delta+D)}$ . Setting $\Theta=\theta C_{1}(s)C_{2}(s)$ , this is equivalent to the following

\mathcal{P}_{H}(\omega,o,o)\mu_{y}(\Omega_{x}^{F}(y,r))=\sum_{h\in\Omega_{x}^{F}(y,r)}\mathrm{e}^{-\omega d(y,ho)}\geq\Theta\sum_{h\in\Omega_{x}^{F}(y,r)}\mathrm{e}^{-\omega d(y,ho)}=\mathcal{P}_{H}(\omega,o,o)\|\mu_{y}\|

Hence, the shadow lemma is proved. ∎

8. Cogrowth tightness of confined subgroups

We continue the setup of §7, but with the first item replaced with a stronger assumption

•

The proper action $\Gamma\curvearrowright\mathrm{X}$ has purely exponential growth.

We first give a uniform upper bound for all conjugates of a confined subgroup $H$ .

Lemma 8.1.

There exist constants $C,\Delta>0$ so that the following holds

\sharp(gHg^{-1}\cap A_{\Gamma}(o,n,\Delta))\leq C\mathrm{e}^{n\omega_{H}}

for any $g\in G$ .

Proof.

By Lemma 5.9, there exists a finite set of contracting elements $\tilde{F}$ such that any $gHg^{-1}$ contains a set, denoted by $F_{g}$ , of three pairwise independent contracting elements from $\tilde{F}$ . Applying Lemma 2.10 with $F_{g}$ , the conclusion follows by the same argument as in [71, Proposition 5.2], where $C$ does not depend on $g$ since there are only finitely many choices of $F_{g}\subseteq\tilde{F}$ independent of $g$ . We include the sketch below and refer to [71, Proposition 5.2] for full details.

Set $S_{n}:=gHg^{-1}\cap A_{\Gamma}(o,n,\Delta)$ . The idea is to prove the following for any $n,m\geq 1$

\sharp S_{n}\sharp S_{m}\leq\theta\sum_{-k\leq j\leq k}\sharp S(n+m+j)

where $\theta,k$ do not depend on $g$ . This is proved by considering the map

	$\displaystyle\pi:\quad S_{n}\times S_{m}$	$\displaystyle\quad\longrightarrow\quad gHg^{-1}$
	$\displaystyle(a,b)$	$\displaystyle\quad\longmapsto\quad afb$

where $f\in F_{g}$ is chosen by Lemma 2.10. In particular, $afb$ labels admissible path so $|d(o,afb)-d(o,ao)-d(o,bo)\leq k$ where $k$ depends on $d(o,fo)$ . We proved there (also see Lemma 2.12) that $\pi$ fails to be injective, only if $d(ao,bo)$ is greater than a constant depending only on $F_{g}$ . This constant determines the value of $\theta$ , so the proof of the above inequality is proved. ∎

Lemma 8.2.

There exist constants $C,\Delta>0$ with the following property. For any $g\in\Gamma$ and any $n\geq 1$ , we have

\sharp(gHg^{-1}\cap A_{\Gamma}(o,n,\Delta))\geq C\mathrm{e}^{n\omega_{\Gamma}/2}

Proof.

Let us consider the conjugate $g_{0}Hg_{0}^{-1}$ for given $g_{0}\in\Gamma$ . Define a map as follows

	$\displaystyle\pi_{g_{0}}:\quad A_{\Gamma}(g_{0}^{-1}o,n,\Delta)$	$\displaystyle\quad\longrightarrow\quad g_{0}Hg_{0}^{-1}$
	$\displaystyle g$	$\displaystyle\quad\longmapsto\quad g_{0}gfp(g_{0}gf)^{-1}$

where $f$ and $p$ are chosen for $g_{0}g$ according to Lemma 5.8. Hence, $|d(o,\pi_{g_{0}}(g)o)-2d(o,g_{0}go)|\leq D$ so we obtain $\pi_{g_{0}}(g)\in A_{\Gamma}(o,2n,\Delta^{\prime})$ where $\Delta^{\prime}:=2\Delta+D$ .

We shall prove that $\pi_{g_{0}}$ is uniformly finite to one. That is to say, there exists a constant $N>1$ independent of $g_{0}$ and $g\in H$ so that $\pi_{g_{0}}^{-1}(g)$ contains at most $N$ elements.

Indeed, if $\pi_{g_{0}}(g)=\pi_{g_{0}}(g^{\prime})$ then $g_{0}gfp(g_{0}gf)^{-1}=g_{0}g^{\prime}fp(g_{0}g^{\prime}f)^{-1}$ . According to Lemma 5.8, the words $(g_{0}g,f,p,f^{-1},(g_{0}g)^{-1})$ and $(g_{0}g^{\prime},f,p,f^{-1},(g_{0}g^{\prime})^{-1})$ label respectively two $(L,\tau)$ –admissible paths with the same endpoints. Thus, any geodesic $\alpha$ with the same endpoints $r$ –fellow travels them, so $g_{0}go$ and $g_{0}g^{\prime}o$ have a distance at most $r$ to $\alpha$ , where $r$ is given by Proposition 2.9. For $g,g^{\prime}\in A_{\Gamma}(g_{0}^{-1}o,n,\Delta)$ , we have $|d(o,g_{0}go)-d(o,g_{0}g^{\prime}o)|\leq 2\Delta$ . Hence, the $r$ –closeness then implies that $d(g_{0}go,g_{0}g^{\prime}o)\leq 4r+2\Delta$ . Thus, if $d(go,g^{\prime}o)$ is larger than a constant $R=R(r,\Delta)$ given by Lemma 2.12, we would obtain a contradiction. The finite number $N:=\sharp\{go:d(o,go)\leq R\}$ is hence the desired upper bound on the preimage of $\pi_{g_{0}}$ , which is clearly uniform independent of $g_{0}\in\Gamma$ .

To conclude the proof, since $\Gamma\curvearrowright\mathrm{X}$ is assumed to have purely exponential growth, we have $\sharp A_{\Gamma}(o,n,\Delta)\geq C_{0}\mathrm{e}^{n\omega_{\Gamma}}$ for some $C_{0},\Delta$ . Setting $C=C_{0}/N$ , we have $\sharp(gHg^{-1}\cap A_{\Gamma}(o,n,\Delta^{\prime}))\geq C\mathrm{e}^{n\omega_{\Gamma}/2}$ . The lemma is proved. ∎

An immediate corollary is as follows. If $\Gamma\curvearrowright\mathrm{X}$ is SCC, this could be also derived from Theorems 1.11 and 1.10, for more general confined subgroups with compact confining subsets. See Corollary 1.12.

Corollary 8.3.

The Poincaré series associated to $H$ diverges at $s=\omega_{\Gamma}/2:$

\mathcal{P}_{H}(\omega_{\Gamma}/2,o,o)=\infty

In particular, $\omega_{H}\geq\omega_{\Gamma}/2$ .

Here is another corollary from the proof. For any $g\in G$ , there exists $p_{g}\in P$ such that $gp_{g}g^{-1}\in H$ . This defines a map as follows:

\pi_{1}:g\in G\longmapsto gp_{g}g^{-1}

Lemma 8.4.

There exists a subset $K\subseteq\Gamma$ , on which the above map $\pi_{1}$ is injective, so that $\sum_{g\in K}\mathrm{e}^{-\omega_{\Gamma}d(o,go)}=\infty$ .

Proof.

The above defined map $\pi_{1}$ is exactly the one $\pi_{g_{0}}$ for $g_{0}=1$ in the proof of Lemma 8.2. As $\pi_{g_{0}}$ is uniformly finite to one and $\Gamma\curvearrowright\mathrm{X}$ is of divergent type, we can then find $K$ with the desired divergence property. ∎

Lemma 8.5.

Suppose that $\omega_{H}=\omega_{\Gamma}/2$ . Then for any $s>\omega_{H}$ and $g\in\Gamma$ , we have

\sum_{h\in H}\mathrm{e}^{-sd(o,ghg^{-1}o)}\asymp\sum_{h\in H}\mathrm{e}^{-sd(o,ho)}

where the implicit constant does not depend on $g$ .

Proof.

We can write for any $g\in\Gamma$ and any $s>\omega_{H}$ ,

\sum_{h\in H}\mathrm{e}^{-sd(o,ghg^{-1}o)}\asymp\sum_{n\geq 1}\sharp(gHg^{-1}\cap A_{\Gamma}(o,n,\Delta))\mathrm{e}^{-sn}

where the implicit constant depends on the width $\Delta$ of the annulus, but is independent of $g$ .

As $H$ contains contracting elements by Lemma 5.10, we have the upper bound by Lemma 8.1

C\mathrm{e}^{n\omega_{H}}\leq\sharp(gHg^{-1}\cap A_{\Gamma}(o,n,\Delta))\leq C^{\prime}\mathrm{e}^{n\omega_{H}}

where the lower bound is given by Lemma 8.2. The conclusion follows by substituting the growth estimates into the above series. ∎

8.1. Confined subgroups of divergence type

We say that a subset intersects trivially a subgroup if their intersection is contained in the trivial subgroup.

Proposition 8.6.

If a discrete group $H<\mathrm{Isom}(\mathrm{X})$ of divergence type is confined by $\Gamma$ with a finite confining subset $P$ that intersects trivially with $E(\Gamma)$ , then $\omega_{H}>\omega_{\Gamma}/2$ .

The proof presented here is similar to the proof of [72, Theorem 9.1(Case 1)], where $H$ is assumed to be normal in $\Gamma$ . We emphasize that $H$ is not necessarily to be contained in $\Gamma$ . We need some preparation before starting the proof.

Fix $go\in\Gamma o$ . Let $\{\mu_{x}^{go}\}_{x\in\mathrm{X}}$ be a PS measure on the horofunction boundary $\partial_{h}{\mathrm{X}}$ , which is an accumulation point of $\{\mu_{x}^{s,o}\}$ in (18) supported on $Hgo$ for a sequence $s\searrow\omega_{H}$ . This is a $\omega_{H}$ –dimensional $H$ –equivariant conformal density.

As $H\curvearrowright\mathrm{X}$ is of divergence type, Lemma 2.37 implies that push forwarding $\{\mu_{x}^{go}\}_{x\in\mathrm{X}}$ for different $go\in\Gamma o$ almost gives the unique quasi-conformal density on the reduced horofunction boundary. That is to say, there are absolutely continuous with respect to each other, with uniformly bounded derivatives. Now, the Patterson’s construction gives

\forall x\in\mathrm{X}:\quad\|\mu_{x}^{s,go}\|=\lim_{s\to\omega_{H}+}\frac{\mathcal{P}_{H}(s,x,go)}{\mathcal{P}_{H}(s,o,go)},\quad\|\mu_{x}^{s,o}\|=\lim_{s\to\omega_{H}+}\frac{\mathcal{P}_{H}(s,x,o)}{\mathcal{P}_{H}(s,o,o)}.

Recall that $H^{g}=g^{-1}Hg$ . From the definition of Poincaré series (15), we have

\forall s>\omega_{H}:\quad\frac{\mathcal{P}_{H}(s,go,go)}{\mathcal{P}_{H}(s,o,go)}=\Bigg{[}\frac{\mathcal{P}_{H}(s,go,o)}{\mathcal{P}_{H^{g}}(s,o,o)}\Bigg{]}^{-1}

On account of Lemma 2.37, we fix in the following statement, a PS measure $\{\mu_{x}\}_{x\in\mathrm{X}}$ on the horofunction boundary $\partial_{h}{\mathrm{X}}$ . This is a limit point of $\{\mu_{x}^{s,o}\}$ in (18) supported on $Ho$ for a sequence $s\searrow\omega_{H}$ .

Lemma 8.7.

If $\omega_{H}=\omega_{\Gamma}/2$ , then there exists a constant $M>0$ such that

(33)

\forall x\in\Gamma o:\quad\|\mu_{x}\|\leq M

Proof.

By Lemma 8.5, if $\omega_{H}=\omega_{\Gamma}/2$ , then $\mathcal{P}_{H^{g}}(s,o,o)\asymp\mathcal{P}_{H}(s,o,o)$ for $s>\omega_{H}$ . Hence, we have

\forall s>\omega_{H}:\quad\frac{\mathcal{P}_{H}(s,go,go)}{\mathcal{P}_{H}(s,o,go)}\asymp\Bigg{[}\frac{\mathcal{P}_{H}(s,go,o)}{\mathcal{P}_{H}(s,o,o)}\Bigg{]}^{-1}

where the implicit constant does not depend on $g$ .

For given $go\in\Gamma o$ , let us take the same sequence of $s$ (depending on $go$ ) so that $\{\mu_{x}^{s,go}\}_{x\in\mathrm{X}}$ and $\{\mu_{x}^{s,o}\}_{x\in\mathrm{X}}$ converge to an $H$ –conformal density $\{\mu_{x}^{go}\}_{x\in\mathrm{X}}$ and $H$ –conformal density $\{\mu_{x}^{o}\}_{x\in\mathrm{X}}$ respectively. The above relation then gives

(34)

\|\mu_{go}^{go}\|\asymp\|\mu_{go}^{o}\|^{-1}

Push forward the limiting measures $\{\mu_{x}^{o}\}_{x\in\mathrm{X}}$ and $\{\mu_{x}^{go}\}_{x\in\mathrm{X}}$ to the quasi-conformal densities on the quotient $[\partial_{m}H]$ , which we keep by the same notation. The constant $\lambda$ in Definition 2.24 is universal for any $\{\mu_{x}^{go}\}_{x\in\mathrm{X}}$ with $go\in\Gamma o$ , as the difference of two horofunctions in the same locus of a Myrberg point is universal. Thus, Lemma 2.37 gives a constant $M_{0}=M(\lambda)$ independent of $go\in\Gamma o$ :

M^{-1}\|\mu_{go}^{go}\|\leq\|\mu_{go}^{o}\|\leq M\|\mu_{go}^{go}\|

so the relation (34) implies $\|\mu_{go}^{o}\|^{2}\leq M$ .

Recall that $\{\mu_{x}\}$ and $\{\mu_{x}^{o}\}$ may be different limit points of $\{\mu_{x}^{s,o}\}_{x\in\mathrm{X}}$ , where $\{\mu_{x}\}$ is the PS measure fixed at the beginning of the proof. However, applying Lemma 2.37 again gives $\|\mu_{x}\|\asymp_{\lambda}\|\mu_{x}^{o}\|$ for any $x\in\mathrm{X}$ and thus for $x=go$ . Combinning with the above bound $\|\mu_{go}^{o}\|^{2}\leq M$ , we obtain the desired upper bound on $\|\mu_{go}\|$ in (33) depending only on $\lambda,M_{0}$ . The proof is completed. ∎

We now recall the following result, which says that a boundary point lies in a uniformly finite many shadows from a fixed annulus.

Lemma 8.8.

[72, Lemma. 6.8] For given $\Delta>0$ , there exists $N=N(\Delta)$ such that for every $n\geq 1$ , any $\xi\in\partial{\mathrm{X}}$ is contained in at most $N$ shadows $\Pi_{o}^{F}(v,r)$ where $v\in A_{\Gamma}(o,n,\Delta)$ .

We are ready to complete the proof of Proposition 8.6.

Proof of Proposition 8.6.

We assume $\omega_{H}=\omega_{\Gamma}/2$ by way of contradiction, so Lemma 8.7 could apply. With the Shadow Principal 7.1, we obtain

\mu_{o}(\Pi_{o}^{F}(go,r))\asymp\mathrm{e}^{-\omega_{H}d(o,go)}

for any $go\in\Gamma o$ . The same argument as in [72, Prop. 6.6] proves $\omega_{H}\geq\omega_{\Gamma}$ . Indeed, by Lemma 8.8, that every point $\xi\in\partial_{h}{\mathrm{X}}$ are contained in at most $N_{0}$ shadows for some uniform $N_{0}>0$ . Hence,

\sum_{v\in A_{\Gamma}(o,n,\Delta)}\mu_{o}(\Pi_{o}^{F}(v,r))\leq N_{0}\mu_{o}(\partial_{h}{\mathrm{X}})\leq N_{0}

which by the Shadow Lemma 2.31 gives $\sharp A_{\Gamma}(o,n,\Delta)\prec\mathrm{e}^{n\omega_{H}}$ , and thus $\omega_{\Gamma}\leq\omega_{H}$ gives a contradiction. ∎

8.2. Completion of proof of Theorem 1.13

First of all, the non-strict inequality $\omega_{H}\geq\frac{\omega_{G}}{2}$ follows from Corollary 8.3. The difficulty is the strict part of this inequality.

If $H\curvearrowright\mathrm{X}$ is of divergence type, then the conclusion follows by Proposition 8.6. If $H\curvearrowright\mathrm{X}$ is of convergence type, then

\sum_{h\in H}\mathrm{e}^{-\omega_{G}d(o,ho)/2}<\infty

and assume for the contradiction that $\omega_{H}=\frac{\omega_{G}}{2}$ . Let $K$ be given by Lemma 8.4, so we obtain that

\sum_{h\in H}\mathrm{e}^{-\omega_{G}d(o,ho)/2}\geq\sum_{g\in K}\mathrm{e}^{-\omega_{G}d(o,gp_{g}g^{-1}o)/2}\succ\sum_{g\in K}\mathrm{e}^{-\omega_{G}d(o,go)}

This contradicts the divergence action of $G\curvearrowright\mathrm{X}$ . This completes the proof of Theorem 1.13.

9. Growth and co-growth Inequality

Under the setup of §7, we now prove Theorem 1.15 following closely Coulon’s argument in [19].

Again, $H$ is only assumed to be confined by $\Gamma$ , but not necessarily contained in $\Gamma$ . Denote $\Gamma/H=\{Hg:g\in\Gamma\}$ the collection of right $H$ –cosets with representatives in $\Gamma$ . Note that $H$ and $Hg$ may be not necessarily contained entirely in $\Gamma$ . Consider the Hilbert space $\mathcal{H}=\ell^{2}(\Gamma/H)$ with over the coset space $\Gamma/H$ .

For each $t>0$ , define

	$\displaystyle\varphi_{t}:\quad\Gamma/H\quad\longrightarrow$	$\displaystyle\quad\mathbb{R}$
	$\displaystyle Hg\quad\longmapsto$	$\displaystyle\quad\mathrm{e}^{-td(Ho,Hgo)}$

Recall that $\omega_{\Gamma/H}$ is the convergence radius of the series

\sum_{g\in\Gamma}\mathrm{e}^{-td(Ho,Hgo)}

Thus, if $t>\omega_{\Gamma/H}/2$ then $\varphi_{t}\in\mathcal{H}$ ; if $t<\omega_{\Gamma/H}/2$ then $\varphi_{t}\notin\mathcal{H}$ . For each $s>0$ , define

	$\displaystyle\phi_{s}:\quad\Gamma/H\quad\longrightarrow$	$\displaystyle\quad\mathbb{R}$
	$\displaystyle Hg\quad\longmapsto$	$\displaystyle\quad\sum_{h\in H}\mathrm{e}^{-sd(o,hgo)}$

We now prove the following.

Lemma 9.1.

For any $s>\omega_{G}$ , we have $\phi_{s}\in\mathcal{H}$ .

Proof.

Consider the PS-measure $\mu_{x}^{s}$ for $s>\omega_{H}$ supported on $Ho$ :

\mu^{s}_{x}=\frac{1}{\mathcal{P}_{H}(s,o,o)}\sum\limits_{h\in H}e^{-sd(x,ho)}\cdot{\mbox{Dirac}}{(ho)}.

We compute the norm

(35)	$\displaystyle\\|\phi_{s}\\|^{2}$	$\displaystyle=\sum_{\underset{g_{1},g_{2}\in\Gamma}{Hg_{1}=Hg_{2}}}\mathrm{e}^{-s(d(o,g_{1}o)+d(o,g_{2}o))}$
		$\displaystyle=\sum_{g\in\Gamma}\mathrm{e}^{-sd(o,go)}\left(\sum_{h\in H}\mathrm{e}^{-sd(go,ho)}\right)$
		$\displaystyle=\mathcal{P}_{G}(s,o,o)\sum_{g\in\Gamma}\mathrm{e}^{-sd(o,go)}\\|\mu_{go}^{s}\\|$

By Shadow Principle 7.3, we have

\|\mu_{go}^{s}\|\mathrm{e}^{-sd(o,go)}\prec\mu_{o}^{s}(\Omega_{o}^{F}(go,r))

Given $n\geq 1$ , any element in the following set

\Omega(o,n)=\{h\in G:d(o,ho)>n\}

is contained at most $N$ members from the family of cones $\{\Omega_{o}^{F}(go,r):g\in A_{\Gamma}(o,n,\Delta)\}$ , where $N$ is a uniform number depending on $r,\Delta$ .

It follows that

\sum_{g\in A(o,n,\Delta)}\|\mu_{go}^{s}\|\mathrm{e}^{-sd(o,go)}\prec\sum_{g\in A(o,n,\Delta)}\mu_{o}^{s}(\Omega_{o}^{F}(go,r))\prec\mu_{o}^{s}(\Omega(o,n))

From (35), we thus obtain

	$\displaystyle\\|\phi_{s}\\|^{2}$	$\displaystyle\prec\sum_{n\geq 1}\left(\sum_{h\in\Omega(o,n)}\mathrm{e}^{-sd(o,ho)}\right)$
		$\displaystyle\prec\sum_{h\in\Omega(o,n)}d(o,ho)\mathrm{e}^{-sd(o,ho)}.$

The last term is the derivative of the Poincaré series $\mathcal{P}_{H}(s,o,o)$ , so is finite for $s>\omega_{H}$ . ∎

Let $s>\omega_{H}$ and $t>\omega_{\Gamma/H}/2$ . Recall that $\mathcal{P}_{\Gamma}(s+t,o,o)=\sum_{g\in\Gamma}\mathrm{e}^{-(s+t)d(o,go)}$ . Note that $\Gamma$ may be properly contained in the union of $Hg$ over $g\in\Gamma$ . Summing up the elements of $\Gamma$ in the same $Hg$ and noting $d(Ho,Hgo)\leq d(o,Hgo)$ , we have

\mathcal{P}_{\Gamma}(s+t,o,o)\leq\sum_{Hg\in\Gamma/H}\left(\mathrm{e}^{-sd(Ho,Hgo)}\sum_{h\in H}\mathrm{e}^{-td(o,hgo)}\right)

The Cauchy-Schwartz inequality gives the finiteness on the scalar product of $\phi_{s}$ and $\varphi_{t}$ :

	$\displaystyle(\phi_{s},\varphi_{r})$	$\displaystyle=\sum_{Hg\in\Gamma/H}\left(\mathrm{e}^{-td(Ho,Hgo)}\sum_{h\in G}\mathrm{e}^{-td(o,hgo)}\right)$
		$\displaystyle\leq\\|\phi_{s}\\|^{2}\\|\varphi_{t}\\|^{2}<\infty$

This implies that $s+t\geq\omega_{H}$ and thus $\omega_{H}+\omega_{\Gamma/H}/2\geq\omega_{\Gamma}$ . Theorem 1.15 is proved.

Part III More about Hopf decomposition, and a relation with quotient growth

10. Characterization of maximal quotient growth

In this section, we relate the conservative/dissipative boundary actions to the growth of quotient spaces. The main result is a dichotomy of quotient growth for any subgroup in a hyperbolic group, where the slower growth is equivalent to the conservative action on the Gromov boundary.

We start with a general development on the Dirichlet domain and its relation with small horospheric limit set.

10.1. Dirichlet domain and small horospheric limit set

The construction of Dirichlet domain tessellates the real hyperbolic spaces into convex polyhedrons (also known as Voronoi tessellation), so provides an important tool to study Kleinian groups with Poincaré polyhedron theorem. In general, the Dirichlet domain can fail to be convex even in other symmetric spaces. The construction of Dirichlet domain is general and relies purely on the metric; in particular, it can be discussed in the setting of coarse metric geometry. Notably, the Nielsen spanning tree is a Dirichlet domain in disguise in the work on free groups of [35]. We first give some variant of the Dirichlet domain in our setting, which relates to the small horospheric limit set (Definition 2.46). This is a key tool for analyzing the quotient growth.

Assume that $H$ acts properly on a proper geodesic space $\mathrm{X}$ with a contracting element. Fix a basepoint $o\in\mathrm{X}$ . Without loss of generality, we may assume that the stabilizer of $o$ in $H$ is trivial: as we can always embed isometrically $\mathrm{X}$ into a larger one (say, attaching a cone at a point with nontrivial stabilizer to make the finite group action free on the base).

Given a (possibly negative) real number $R$ , let $\mathbf{D}_{R}(o)$ be the set of points $x\in\mathrm{X}$ that are $R$ –closer to $o$ than any point in $Ho$ . Namely, $x\in\mathbf{D}_{R}(o)$ if and only if $d(x,o)\leq d(x,Ho)+R$ . Equivalently, $\mathbf{D}_{R}(o)=\cap_{1\neq h\in H}\overleftarrow{\mathbf{H}}_{R}(o,ho)$ is a countable intersection of $R$ –half spaces defined as follows

\overleftarrow{\mathbf{H}}_{R}(x,y):=\{z\in\mathrm{X}:d(z,x)\leq d(z,y)+R\}

This forms a locally finite family of closed sets (via the same proof of [59, Theorem 6.6.13]), so $\mathbf{D}_{R}(o)$ is a closed subset. It is obvious that $\mathbf{D}_{R}(o)\subseteq\mathbf{D}_{R^{\prime}}(o)$ for $R\leq R^{\prime}$ , and $\mathbf{D}_{-R}(o)\subseteq\mathbf{D}_{0}(o)\subseteq\mathbf{D}_{R}(o)$ for $R>0$ . See Fig. 8 for an illustration of these notions.

We remark that the introduction of $\mathbf{D}_{R}(o)$ with negative $R$ is an essential novelty here, which is crucial in the Claim 2 in proof of Theorem 10.13.

Dirichlet domain

For $R=0$ , $\mathbf{D}(o):=\mathbf{D}_{0}(o)$ is the so-called Dirichlet domain centered at $o$ for the action $H\curvearrowright\mathrm{X}$ . It is a fundamental domain in the following sense: $H\cdot\mathbf{D}(o)=\mathrm{X}$ and $h\cdot\textrm{Int}(\mathbf{D}(o))\cap\textrm{Int}(\mathbf{D}(o))=\emptyset$ for any $h\neq 1$ . Denote by $\pi:\mathrm{X}\to\mathrm{X}/H$ the quotient map, whose quotient topology is induced by the metric $\bar{d}$ . By [59, Theorem 6.6.7], $\mathrm{X}/H$ is homeomorphic to $\pi(\mathbf{D}(o))$ , where the latter is equipped with quotient topology via the restriction $\pi:\mathbf{D}(o)\to\pi(\mathbf{D}(o))$ . See [59, Ch. 6] for relevant discussion.

In a real hyperbolic space, the notion of a bisector is useful in analyzing the Dirichlet domain, as it intersects the convex polyhedron $\mathbf{D}(o)$ in faces. We here adopt a more general version of bisectors. For $R\geq 0$ , set $\mathrm{Bis}(x,y;R):=\{z\in\mathrm{X}:|d(z,x)-d(z,y)|\leq R\}$ , and then $\overleftarrow{\mathbf{H}}_{R}(x,y)=\overleftarrow{\mathbf{H}}_{0}(x,y)\cup\mathrm{Bis}(x,y;R)$ .

Examples 10.1.

Here are two examples to clarify the notion of bisectors.

(1)

In trees, it is readily checked that $\overleftarrow{\mathbf{H}}_{R}(x,y)$ is not contained in $N_{T}(\overleftarrow{\mathbf{H}}_{0}(x,y))$ for any $T>0$ . Indeed, the limit set of the former set properly contains that of the latter set in the end boundary.
(2)

In a simply connected negatively pinched Riemannian manifold, it can be shown that $\mathrm{Bis}(x,y;R)$ has finite Hausdorff distance (depending on $R$ ) to $\mathrm{Bis}(x,y;0)$ (using crucially the lower bound of curvature, i.e. the fatness of the comparison triangle). It follows that $\mathrm{Bis}(x,y;R)$ is quasiconvex and its limit set remains the same for any $R\geq 0$ .

Figure 8. Illustrating

R

–half spaces for possible

R

and their resulted Dirichlet domains

Let $S^{+R}$ denote the closed $R$ –neighborhood of a subset $S$ in $\mathrm{X}$ .

Lemma 10.2.

The following holds for any real number $R\in\mathbb{R}$ .

(1)

$\mathbf{D}_{R}(o)$ is star-shaped at $o$ : any geodesic $[o,x]$ for $x\in\mathbf{D}_{R}(o)$ is contained in $\mathbf{D}_{R}(o)$ .
(2)

If $R\geq 0$ , then $\mathbf{D}^{+R}(o))\subseteq\mathbf{D}_{2R}(o)$ .

Proof.

(1). For any $x\in\mathbf{D}_{R}(o)$ , we wish to prove that any geodesic $[x,o]$ is contained in $\mathbf{D}_{R}(o)$ . Indeed, if $y\in(x,o)$ is not in $\mathbf{D}_{R}(o)$ , there exists $h\in H$ so that $d(y,ho)<d(y,o)+R$ . This implies $d(x,ho)\leq d(x,y)+d(y,ho)<d(x,o)+R$ , a contradiction with $x\in\mathbf{D}_{R}(o)$ .

(2). If $x\in\mathbf{D}^{+R}(o)$ , then for some $y\in\mathbf{D}(o)$ , we have $d(x,y)\leq R$ and $d(y,o)\leq d(y,Ho)$ . So $d(x,o)\leq d(y,Ho)+R\leq d(x,Ho)+2R$ , i.e.: $x\in\mathbf{D}_{2R}(o)$ . ∎

We now fix a convergence boundary $\partial{\mathrm{X}}$ for $\mathrm{X}$ , denote by $\Lambda(S)$ the set of accumulations points of $S$ in $\partial{\mathrm{X}}$ . By Definition 2.13(B), we have $[\Lambda(S^{+R})]=[\Lambda(S)]$ .

The main result of this subsection is the following.

Theorem 10.3.

For any large $R>0$ ,

(1)

$[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]\setminus\Lambda^{\mathrm{Hor}}{(Ho)}=H\cdot\left([\Lambda\mathbf{D}_{R}(o)]\cap[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]\right).$
(2)

$\Lambda^{\mathrm{hor}}{(Ho)}\cap[\Lambda\mathbf{D}_{R}(o)]=\emptyset$ .

It is worth noting the following much simplified statement in hyperbolic or CAT(0) spaces. We remark that the proof in this case could be quite short and straightforward (cf. [51, Cor. 2.14]). We recommend the readers to figure out the proof themselves, instead of reading the following one which deals mainly with general metric spaces.

Theorem 10.4.

Let $\mathrm{X}$ be Gromov hyperbolic with Gromov boundary $\partial{\mathrm{X}}$ or CAT(0) with visual boundary $\partial{\mathrm{X}}$ . Then for any large $R>0$ .

(1)

$\partial{\mathrm{X}}\setminus\Lambda^{\mathrm{Hor}}{(Ho)}=H\cdot\Lambda\mathbf{D}_{R}(o).$
(2)

$\Lambda^{\mathrm{hor}}{(Ho)}\cap\Lambda\mathbf{D}_{R}(o)=\emptyset$ .

The proof is achieved by a series of elementary lemmas.

Recall that a point $\xi$ is visual if a geodesic ray $\gamma$ originating at $o$ terminates at $[\xi]$ , i.e. all the accumulation points of $\gamma$ are contained in $[\xi]$ . We emphasize that $\xi$ may not be an accumulation point of $\gamma$ , so it is necessary to make statements on $[\cdot]$ –classes rather than boundary points. Any $(r,F)$ –conical point is visual by Lemma 2.35.

Remark 10.5.

The assumption $\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ in the next three results is used to guarantee the following. If $x_{n}\to\xi$ , then any limiting geodesic ray of (a subsequence of) $[o,x_{n}]$ accumulates into $[\xi]$ . If $\mathrm{X}$ is hyperbolic or CAT(0), this fact holds for any $\xi\in\partial{\mathrm{X}}$ with $\partial{\mathrm{X}}$ being Gromov boundary or visual boundary. Therefore, $\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ could be replaced with $\partial{\mathrm{X}}$ in these setups.

Recall the definition of horoballs in (2.22). The assertion (1) in Theorem 10.3 is proved by the following.

Lemma 10.6.

There exists a constant $R=R(F)>0$ with the following property. Let $\xi\in[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]\setminus\Lambda^{\mathrm{Hor}}{(Ho)}$ be not a big horospheric limit point. Then there exist some $h\in H$ and a geodesic $[ho,\xi]$ which is contained in $\mathbf{D}_{R}(ho)$ .

Proof.

According to definition, if $\xi\notin\Lambda^{\mathrm{Hor}}{(Ho)}$ , some horoball centered at $[\xi]$ contains only finitely many elements from $Ho$ , so $L=\min\{B_{[\xi]}(ho,o):h\in H\}$ is a finite value. Moreover, the horoball $\mathcal{HB}([\xi],L)$ contains some $ho\in Ho$ , but $\mathcal{B}_{[\xi]}(ho,h^{\prime}o)\leq 0$ for any $h^{\prime}o\in Ho$ . Let $C$ be the common contracting constant for elements in $F$ . Note that $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ is a subset of $\mathcal{C}^{\mathrm{hor}}_{20C}$ by Lemma 2.35. For any sequence of $x_{n}\to\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ , we obtain from (8) for $n\gg 0$ :

B_{x_{n}}(h_{n}o,h_{n}^{\prime}o)=d(x_{n},ho)-d(x_{n},h^{\prime}o)\leq 21C

That is, setting $R=21C$ , $x_{n}\in\overleftarrow{\mathbf{H}}_{R}(ho,h^{\prime}o)$ for any $h^{\prime}\in H$ , so we obtain $x_{n}\in\mathbf{D}_{R}(o)$ by definition. See Fig. 9. Now, take a geodesic ray $\gamma$ starting at $ho$ and ending at $[\xi]$ by Lemma 2.35 (cf. Remark 10.5). If $x_{n}$ is taken on $\gamma$ tending to $[\xi]$ , the star-sharpedness implies $\gamma\subseteq\mathbf{D}_{R}(o)$ by Lemma 10.2. Thus $\xi\in[\Lambda\mathbf{D}_{R}(ho)]$ . ∎

Figure 9. Proof of Lemma 10.6

The following preparatory result is required in proving the assertion (2) in Theorem 10.3. This follows easily in hyperbolic or CAT(0) spaces, as explained in Remark 10.5.

Lemma 10.7.

Let $\xi\in[\Lambda\mathbf{D}_{R}(o)]\cap[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ for some $R>0$ . Then $\mathbf{D}_{R}(o)$ contains a geodesic ray ending at $[\xi]$ starting at $o$ .

Proof.

By [72, Lemma 4.4], such a geodesic ray exists for any $\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ . We briefly recall the proof, and then explain $\gamma$ can be taken into $\mathbf{D}_{R}(o)$ by using the star-shaped $\mathbf{D}_{R}(o)$ . First, a conical point $\xi\in\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}$ by definition lies in infinitely many partial shadows $\Pi_{o}^{F}(g_{n}o,r)$ for $g_{n}\in\Gamma$ . By the proof of [72, Lemma 4.4], we can find a sequence of $y_{n}\in\Pi_{o}^{F}(g_{n}o,r)$ tending to $\xi$ , so that $[o,y_{n}]$ intersects $N_{C}(g_{n}\mathrm{Ax}(f))$ for a common $f\in F$ in a diameter comparable with $d(o,fo)$ (as $\sharp F=3$ ). By Ascoli-Arzela Lemma, the limiting geodesic ray of (a subsequence of) $[o,y_{n}]$ tends to $[\xi]$ by Definition 2.13(B).

Now take $x_{n}\in\mathbf{D}_{R}(o)$ tending to $[\xi]$ . If $x_{n}\in\Pi_{o}^{F}(g_{n}o,r)$ , then $[o,x_{n}]\subseteq\mathbf{D}_{R}(o)$ converges locally uniformly to a geodesic ray $\gamma\subseteq\mathbf{D}_{R}(o)$ by Lemma 10.2. In the general case, if $d(o,fo)$ is sufficiently large, the $C$ –contracting property of $g_{n}\mathrm{Ax}(f)$ implies that $[o,x_{n}]$ intersects $N_{C}(g_{n}\mathrm{Ax}(f))$ in a diameter comparable with $d(o,fo)$ . Hence, a subsequence of $[o,x_{n}]$ converges locally uniformly to a geodesic ray $\alpha$ ending at $[\xi]$ . As we wanted, $\alpha$ is contained in the star-shaped $D_{R}(o)$ . ∎

We now prove the assertion (2) in Theorem 10.3

Lemma 10.8.

Let $\xi\in[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]\cap[\Lambda\mathbf{D}_{R}(o)]$ for some $R>0$ . Then $\xi\notin\Lambda^{\mathrm{hor}}{(Ho)}$ .

Proof.

By Lemma 10.7, take a geodesic ray $\gamma\subseteq\mathbf{D}_{R}(o)$ starting at $o$ and terminating at $[\xi]$ .

By way of contradiction, assume that $\xi\in\Lambda^{\mathrm{hor}}{(Ho)}$ is a small horospheric limit point. Hence, any horoball $\mathcal{HB}([\xi])$ centered at $[\xi]$ contains a sequence of $h_{n}o\in Go$ tending to $\xi$ . That is, $B_{\xi}(h_{n}o,o)\to-\infty$ as $n\to\infty$ . Note that the Busemann cocycle associated to $\gamma$ differs from $B_{[\xi]}(h_{n}o,o)$ up to a uniform error. This implies that $d(x,h_{n}o)-d(x,o)\to-\infty$ as $x\in\gamma$ tends to $[\xi]$ . Consequently, the definition of $\mathbf{D}_{R}(o)$ shows that $x\notin\mathbf{D}_{R}(o)$ for any fixed $R$ . This contradicts the choice of $\gamma$ in $\mathbf{D}_{R}(o)$ . ∎

Finally, the proof of Theorem 10.4 follows from Lemma 10.6 and Lemma 10.8, where $[\Lambda^{\textrm{con}}_{\textrm{r,F}}{(\Gamma o)}]$ could be replaced with $\partial{\mathrm{X}}$ by Remark 10.5.

10.2. Big and small horospheric limit set

In this and next subsections, assume that

•

The proper action $\Gamma\curvearrowright\mathrm{X}$ is cocompact on a proper hyperbolic space $(\mathrm{X},d)$ , which is compactified with Gromov boundary $\partial{\mathrm{X}}$ endowed with maximal partition (i.e. each $[\cdot]$ -class is singleton, so $[]$ is omitted in what follows).
•

Let $\{\mu_{x}:x\in\mathrm{X}\}$ be the family of PS measures on $\Lambda(\Gamma o)=\partial{\mathrm{X}}$ constructed from $\Gamma\curvearrowright\mathrm{X}$ . Then $\mu_{o}$ is a doubling measure without atoms; indeed it is the Hausdorff measure with respect to visual metric at the correct dimension (see [18]). Any subgroup $H<\mathrm{Isom}(\mathrm{X})$ preserves the measure class of $\mu_{o}$ by Lemma 4.1.

Let a countable group $H$ (possibly not contained in $\Gamma$ ) act properly on $\mathrm{X}$ . We shall prove the following by elaborating on the proof of Sullivan [65] in Kleinian groups (see also [51, Lemma 5.7]).

Theorem 10.9.

$\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)}\setminus\Lambda^{\mathrm{hor}}{(Ho)})=0$ .

The key part is to prove that the set of Garnett points is $\mu_{o}$ –null. Recall that a point $\xi\in\Lambda^{\mathrm{Hor}}{(Ho)}$ is called Garnett point if there exists a unique horoball $\mathcal{HB}(\xi,L)$ (Definition 2.22) at $\xi$ so that

•

The interior of $\mathcal{HB}(\xi,L)$ contains no point from $Ho$ : $B_{\xi}(ho,o)\geq L$ for any $ho\in Ho$ ;
•

any larger horoball contains infinitely many points of $Ho$ : $B_{\xi}(ho,o)\in[L,L+\epsilon]$ for infinitely many $ho\in Ho$ and any $\epsilon<0$ ;

where $B_{\xi}(ho,o)=\limsup_{z\to\xi}[d(z,ho)-d(z,o)]$ . Denote by $\mathbf{Gar}$ the set of Garnett points for $H$ .

We first recall a useful observation of Tukia [67, Appendix].

Lemma 10.10.

The set $A=\Lambda^{\mathrm{Hor}}{(Ho)}\setminus(\Lambda^{\mathrm{hor}}{(Ho)}\cup\mathbf{Gar})$ is $\mu_{o}$ –null.

Proof.

We include a short proof for completeness. Indeed, let $\xi\in\Lambda^{\mathrm{Hor}}{(Ho)}\setminus\Lambda^{\mathrm{hor}}{(Ho)}$ be a non-Garnett point, so there exists a horoball $\mathcal{HB}(\xi)$ whose boundary contains a non-empty finite set of elements in $Ho$ but whose interior contains no point in $Ho$ . Consequently, $A$ is a disjoint union of measurable subsets indexed over finite subsets in $Ho$ , which are invariant under $H$ , so we could choose a measurable fundamental domain for $H\curvearrowright A$ . By Lemma 2.48, $\Lambda^{\mathrm{Hor}}{(Ho)}$ is a subset of the conservative part $\mathbf{Cons}$ , so $\mu_{o}(A)=0$ . ∎

To complete the proof of Theorem 10.9, it suffices to prove that $\mathbf{Gar}$ is a $\mu_{o}$ –null set.

Lemma 10.11.

$\mu_{o}(\mathbf{Gar})=0$ .

Proof.

By way of contradiction, assume that $\mu_{o}(\mathbf{Gar})>0$ .

We define a height function $\mathfrak{h}:\mathbf{Gar}\to\mathbb{R}_{\geq 0}$ . Set $\mathfrak{h}(\xi)=d(o,\mathcal{H}(\xi))$ , where $\mathcal{H}(\xi)$ is the unique horoball centered at $\xi$ whose interior does not contain any point of $Ho$ and any horoball centered at $\xi$ larger than that contains infinitely many points of $ho$ . We verify that $\mathfrak{h}$ is a measurable function. Indeed, if $\xi\in\Lambda\mathrm{Bis}(x,y;\epsilon)$ , then $|B_{\xi}(x,y)|\leq\epsilon$ ; if $|B_{\xi}(x,y)|\leq\epsilon$ , then $\xi\in\Lambda\mathrm{Bis}(x,y;\epsilon+1/n)$ for any $n>1$ . This implies that the set $\xi\in\partial{\mathrm{X}}$ with $|B_{\xi}(x,y)|\leq\epsilon$ is a countable intersection of closed subsets, so is Borel measurable. By definition, $\mathbf{Gar}$ is the limit supremum of such measurable sets. Moreover, the set of $\xi\in\mathbf{Gar}$ with $\mathfrak{h}(\xi)\in(r_{1},r_{2})$ is measurable.

As $\mu_{o}$ is a doubling measure, the Lebesgue density theorem holds: Let $\mathbf{G}$ be the $\mu_{o}$ –full set of density points $\xi\in\mathbf{Gar}$ , so

(1)

$\mathfrak{h}:\mathbf{G}\to\mathbb{R}$ is approximately continuous: $\xi_{n}\in\mathbf{G}\to\xi\in\mathbf{G}$ implies $\mathfrak{h}(\xi_{n})\to\mathfrak{h}(\xi)$ ;
(2)

for any sequence of metric balls $B_{n}\subseteq\partial{\mathrm{X}}$ centered at $\xi\in\mathbf{G}$ with radius tending to 0, we have

$\frac{\mu_{o}(\mathbf{Gar}\cap B_{n})}{\mu_{o}(B_{n})}\to 1.$

As in [65], we are going to find a “forbidden” region $C_{n}\subseteq B_{n}$ , consisting of non-Garnett points, with a definite $\mu_{o}$ –measure. This contradicts the item (2). The co-compact action is crucial to apply Shadow Lemma 2.31: for any $x\in\mathrm{X}$ and any fixed large constant $r$ , we have

(36)

\mu_{o}(\Pi_{o}(x,r))\asymp_{r}\mathrm{e}^{-\omega_{\Gamma}d(o,x)}.

Figure 10. The proof of Lemma 10.11 where

\mathrm{X}

is a tree and

r=0

. The forbidden region

C_{n}

contains no Garnett point: a horoball at

\zeta\in C_{n}

containing

x

must contain

h_{n}o

For $\xi\in\mathbf{Gar}$ , there exists a unique horoball $\mathcal{HB}(\xi,L_{\xi})$ for some real number $L_{\xi}$ , so that $B_{\xi}(o,ho)>L_{\xi}$ for all $h\in H$ , but $L_{\xi}+1>B_{\xi}(o,h_{n}o)>L_{\xi}$ for infinitely many distinct $h_{n}o\in Ho$ . Let $x\in\mathcal{HB}(\xi,L_{\xi})$ be a projection point of $o$ onto $\mathcal{HB}$ , so $\mathfrak{h}(\xi)=d(o,x)$ by definition. As horoballs in hyperbolic spaces are quasi-convex, we have $|L_{\xi}-\mathfrak{h}(\xi)|\leq 100\delta$ , and $d(x,[o,h_{n}o])\leq 100\delta$ .

Denote $L_{n}:=d(x,h_{n}o)$ . Choose $p_{n},q_{n}\in[x,h_{n}o]$ with $d(x,p_{n})=L_{n}/2$ and $d(x,q_{n})=L_{n}/2+D$ for a large constant $D\gg r$ . Set $B_{n}:=\Pi_{o}(p_{n},r)$ and $C_{n}:=\Pi_{o}(q_{n},r)$ . See Fig. 10 for approximate tree configuration of these points.

As $d(x,[o,h_{n}o])\leq 100\delta$ and $|d(x,p_{n})-d(x,q_{n})|\leq D$ , we see that $|d(o,p_{n})-d(o,q_{n})|\leq D+200\delta$ . Shadow estimates (36) implies the existence of a positive constant $c$ depending only on $D$ :

{\mu_{o}(C_{n})}\geq c{\mu_{o}(B_{n})}

We claim that for all $n\gg 0$ , $C_{n}$ consists of non-Garnett points up to a $\mu_{o}$ -null set. If not, take a density point $\zeta\in C_{n}\cap\mathbf{G}$ . Let $\mathcal{HB}(\zeta,L_{\zeta})$ be the unique horoball associated with $\zeta$ , so $|L_{\zeta}-\mathfrak{h}(\zeta)|\leq 100\delta$ follows as above. Let $y\in\mathcal{HB}(\zeta,L_{\zeta})$ so that $d(o,y)=\mathfrak{h}(\zeta)$ . Our goal is to prove $\mathcal{HB}(\zeta,L_{\zeta})$ must contain $h_{n}o$ , contradicting the definition of a Garnett point.

Indeed, the approximate continuity implies $|\mathfrak{h}(\xi)-\mathfrak{h}(\zeta)|=|d(o,x)-d(o,y)|\leq 1$ for $\zeta\in C_{n}\to\xi$ . As $\zeta\in\Pi_{o}(q_{n},r)$ , we have $d(q_{n},[o,\zeta))\leq r$ . If $D\gg r$ , the thin-triangle inequality shows that $d(x,[o,\zeta])\leq 10\delta$ and thus $d(x,y)\leq 100\delta$ . Now, by the choice of $d(x,p_{n})=L_{n}/2$ and $d(h_{n}o,q_{n})=L_{n}/2-D$ , we have $d(y,q_{n})\geq d(x,q_{n})-d(x,y)\geq L_{n}/2+D-100\delta$ , implying

d(h_{n}o,q_{n})\leq L_{n}/2-D\leq d(y,q_{n})-2D+100\delta

Noting $d(q_{n},z)\leq r$ for some $z\in[o,\zeta)$ , we choose $D\gg r$ so that $d(h_{n}o,z)\leq d(y,z)-100\delta$ . Lemma 6.1 implies $h_{n}o\in\mathcal{HB}(\zeta,L_{\zeta})$ . Hence, we conclude that $C_{n}\cap\mathcal{G}=\emptyset$ . Recall that ${\mu_{o}(C_{n})}\geq c{\mu_{o}(B_{n})}$ , this contradicts the property that $\xi$ is a density point as described in (2). The proof of the lemma is complete. ∎

We conclude this subsection with a remark on possible extensions of Theorem 10.9 to mapping class groups.

Remark 10.12.

Consider the (non-cocompact) action of the mapping class group $\Gamma=\mathbf{Mod}(\Sigma_{g})$ on Teichmüller space $\mathrm{X}$ for $g\geq 2$ . The proof of Lemma 10.11 could be adapted to its subgroup $H<\Gamma$ , provided that (36) holds for all $x\in\mathrm{X}$ . (Hyperbolicity could be circumvented by using partial shadows). By the Shadow Lemma 2.31, it is known that (36) holds for a cocompact subset of points. By work of [7], the conformal density $\mu_{x}$ gives the same mass to Thurston boundary $\mathscr{PMF}$ , and the volume growth in Teichmüller space does not depend on the basepoint. This provides positive evidence that (36) holds for all $x\in\mathrm{X}$ .

10.3. Characterizing maximal quotient growth

Let $B(o,n)\subseteq\mathrm{X}$ and $\bar{B}(\pi(o),n)\subseteq\mathrm{X}/H$ denote respectively the $d$ –ball and $\bar{d}$ –ball at $o$ and $\pi(o)$ of radius $n\geq 0$ . Note that $\pi(B(o,n))=\bar{B}(\pi(o),n)$ : $Hgo\in B(\pi(o),n)$ if and only if $d(Ho,Hgo)\leq n$ . We say that $\mathrm{X}/H$ has slower growth than $\mathrm{X}$ if given any maximal $R$ –separated subset $Z$ (resp. $W$ ) of $\mathrm{X}$ (resp. $\mathrm{X}/H$ ) for some $R>0$ , we have

\frac{\sharp W\cap B(\pi o,n)}{\sharp Z\cap B(o,n)}\to 0,\;\text{as }n\to\infty.

This definition is independent of the choice of $Z,W$ and $R$ , as different choice yields comparable ball growth function up to multiplicative constants. By the Shadow Lemma, the notion of slower growth in orbit points could be re-formulated using volume. For example, if $\mathrm{X}$ is a simply connected rank-1 manifold with a co-compact action, $\sharp Z\cap B(o,n)$ is comparable to the volume of $B(o,n)$ . In a non-cocompact action, an important potential example could be given by mapping class groups, if the full shadow principle holds as stated in Remark 10.12.

The following result in Kleinian groups was first obtained in [65, Theorem VI] and later in free groups in [35, Theorem 4.1] and [8, Theorem 7]. Recall from the previous subsection, $\Gamma\curvearrowright\mathrm{X}$ is assumed to be geometric action on a hyperbolic space, and $\mu_{o}$ is the corresponding Hausdorff measure on the Gromov boundary $\partial{\mathrm{X}}$ .

Theorem 10.13.

Assume that $H$ acts properly on a proper hyperbolic space $\mathrm{X}$ . Then the small/big horospheric limit set is $\mu_{o}$ –full, if and only if $\mathrm{X}/H$ grows slower than $\mathrm{X}$ . Moreover, $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})<1$ is equivalent to the following

\sharp\{Hgo:\bar{d}(Ho,Hgo)\leq n,g\in\Gamma\}\asymp\mathrm{e}^{\omega_{\Gamma}n}.

As an immediate corollary, we obtain a characterization of conservative actions.

Corollary 10.14.

Assume that $H$ is torsion-free. Then the action $H\curvearrowright(\partial{\mathrm{X}},\mu_{o})$ is conservative if and only if the action $H\curvearrowright\mathrm{X}$ grows slower than $\Gamma\curvearrowright\mathrm{X}$ .

Proof of Theorem 10.13.

By Theorems 10.9 and 10.3, $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})=\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})=1$ if and only if $\mu_{o}(\Lambda\mathbf{D}_{R}(o))=0$ for any sufficiently large $R>0$ .

Let $\pi:\mathrm{X}\to\mathrm{X}/H$ be the projection. Fix a basepoint $o\in\mathrm{X}$ and consider the orbit $A=\Gamma o$ . Then $N_{R}(A)=\mathrm{X}$ for some $R>0$ as the action $\Gamma\curvearrowright\mathrm{X}$ is co-compact by assumption. The image $\bar{A}=\pi(A)$ is equipped with the metric $\bar{d}(Hx,Hy)=d(Hx,Hy)$ for $x,y\in A$ , and write

A_{\Gamma/H}(o,n,\Delta)=\{Hgo:|d(go,Ho)-n|\leq\Delta\}

According to (3), we have

\omega_{\Gamma/H}=\limsup_{n\to\infty}\frac{\log A_{\Gamma/H}(o,n,\Delta)}{n}

Recall $A_{\Gamma}(o,n,\Delta)=\{go\in\Gamma o:|d(o,go)-n|\leq\Delta\}$ . For each $Hgo\in A_{\Gamma/H}(o,n,\Delta)$ , choose $go\in\Gamma o$ so that $d(go,o)=d(Hgo,Ho)$ . That is, $go$ is a closest point in $Hgo$ to $o$ , so $go$ is an element in $\mathbf{D}(o)$ . Thus, every element in $A_{\Gamma/H}(o,n,\Delta)$ admits a (possibly non-unique) lift in the following intersection:

B_{n}:=A_{\Gamma}(o,n,\Delta)\cap\mathbf{D}(o)

Thus, $\sharp A_{\Gamma/H}(o,n,\Delta)\leq\sharp B_{n}$ .

Claim.

If $\mu_{o}(\Lambda\mathbf{D}_{R}(o))=0$ , then $\sharp B_{n}=o(\mathrm{e}^{\omega_{\Gamma}n})$ and $\sharp A_{\Gamma/H}(o,n,\Delta)=o(\mathrm{e}^{\omega_{\Gamma}n})$

Proof of the Claim.

Define

\Lambda_{n}:=\bigcup_{go\in B_{n}}\Pi_{o}^{F}(go,r).

Let $\Lambda$ be the limit supremum of $\Lambda_{n}$ . By construction, $\Lambda\mathbf{D}_{R}(o)$ coincides with $\Lambda$ . Indeed, any point $\xi\in\Lambda\mathbf{D}_{R}(o)$ , there exists a geodesic ray in $\mathbf{D}_{R}(o)$ ending at $\xi$ . Then we can choose $g_{n}\in B_{n}$ so that $d(g_{n}o,\gamma)\leq R$ . This implies $\Lambda\mathbf{D}_{R}(o)\subseteq\Lambda$ . The converse inclusion follows from the definition of $B_{n}$ . Hence, $\mu_{o}(\Lambda_{n})\to 0$ as $n\to\infty$ .

As the family of shadows $\Pi_{o}^{F}(x,r)$ for $x\in B_{n}$ has uniformly bounded multiplicity, we have by the Shadow Lemma

\mu_{o}(\Lambda_{n})\asymp\sum_{x\in B_{n}}\mu_{o}(\Pi_{o}^{F}(x,r))\asymp\mathrm{e}^{-\omega_{\Gamma}n}\sharp B_{n}

where the implicit constant depends on $r$ . Hence, the claim follows. ∎

Claim.

If $\mu_{o}(\Lambda\mathbf{D}_{R}(o))>0$ , then $\sharp A_{\Gamma/H}(o,n,\Delta)\asymp\mathrm{e}^{\omega_{\Gamma}n}$ .

Proof of the Claim.

Recall that $\mathbf{D}(o)$ is a fundamental domain for $H\curvearrowright\mathrm{X}$ , so any point $x\in\mathrm{X}$ admits at least one translate $hx$ into $\mathbf{D}(o)$ . If $h_{1}x\neq h_{2}x\in\mathbf{D}(o)$ , we have $d(h_{1}x,o)=d(h_{2}x,o)$ and then $x$ lies on the bisector $\mathrm{Bis}(h_{1}^{-1}o,h_{2}^{-1}o;0)$ .

Consider the facet-like set

F(h):=\mathrm{Bis}(o,ho;R)\cap\mathbf{D}_{R}(o)

and the set $\Lambda(h)$ of accumulation points of $F(h)$ in $\partial{\mathrm{X}}$ . By Theorem 10.3, the intersection $\Lambda(h)\cap\Lambda^{\mathrm{Hor}}{(Ho)}$ has no small horospheric limit point, so is $\mu_{o}$ –null by Theorem 10.9.

Note that $\mathbf{D}_{-R}(o)$ for $R>0$ is contained in the interior of $\mathbf{D}(o)$ , where $\pi:\mathrm{X}\to\mathrm{X}/H$ restricts as an injective map. See Fig. 8. By definition, $\Lambda\mathbf{D}_{R}(o)$ is the union of $\Lambda\mathbf{D}_{-R}(o)$ with countably many facet-like sets $F(h)$ , which are $\mu_{o}$ -null, so we obtain $\mu_{o}(\Lambda\mathbf{D}_{-R}(o))=\mu_{o}(\Lambda\mathbf{D}_{R}(o))>0$ .

By injectivity of $\pi:\mathbf{D}_{-R}(o)\to\mathrm{X}/H$ restricted as map, $\hat{B}_{n}:=A_{\Gamma}(o,n,\Delta)\cap\mathbf{D}_{-R}(o)$ injects into $\mathrm{X}/H$ . Consider similarly

\hat{\Lambda}_{n}:=\bigcup_{go\in\hat{B}_{n}}\Pi_{o}^{F}(go,r).

Let $\tilde{\Lambda}$ be the limit of $\tilde{\Lambda}_{m}:=\cup_{m\geq n}\hat{\Lambda}_{m}$ . As the above claim, the star-shapedness of $\mathbf{D}_{-R}(o)$ shows that $\tilde{\Lambda}=\Lambda\mathbf{D}_{-R}(o)$ , and $\tilde{\Lambda}_{m}$ is covered with a uniform multiplicity by the family of $\Pi_{o}^{F}(go,r)$ for ${go\in\hat{B}_{n}}$ .

Consequently, $\mu_{o}(\hat{B}_{n})>c$ for some uniform $c>0$ independent of $n$ , and

\mu_{o}(\hat{B}_{n})\asymp\sum_{x\in\hat{B}_{n}}\mu_{o}(\Pi_{o}(x,r))\asymp\mathrm{e}^{-\omega_{\Gamma}n}\sharp\hat{B}_{n}

We thus obtain $\sharp\hat{B}_{n}\asymp\mathrm{e}^{\omega_{\Gamma}n}$ , and then the conclusion follows by $\sharp\hat{B}_{n}\leq\sharp A_{\Gamma}(o,n,\Delta)$ . ∎

The proof of the theorem is completed by the above two claims. ∎

The first claim actually does not make use of the assumption that $\Gamma\curvearrowright\mathrm{X}$ is cocompact and $\mu_{o}$ is doubling. These two assumptions are only required in Lemma 10.11. So we have.

Corollary 10.15.

Assume that $\Gamma\curvearrowright\mathrm{X}$ is a proper action on a hyperbolic space and $\mu_{x}$ be the $\omega_{\Gamma}$ –dimensional $\Gamma$ –equivariant quasi-conformal density on $\partial{\mathrm{X}}$ . Let $H$ be a subgroup in $\Gamma$ . If $\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})=0$ , then $\sharp A_{\Gamma/H}(o,n,\Delta)=o(\mathrm{e}^{\omega_{\Gamma}n})$ .

11. Subgroups with nontrivial conservative and dissipative components

To complement the part 1, we construct in this section examples with non-trivial Hopf decomposition. Through this section, suppose that

•

$\Gamma$ admits a proper SCC action on $\mathrm{X}$ with contracting elements.
•

Let $\partial{\mathrm{X}}$ be a convergence boundary for $\mathrm{X}$ and $\{\mu_{x}:x\in\mathrm{X}\}$ be a quasi-conformal, $\Gamma$ –equivariant density of dimension $\omega_{\Gamma}$ on $\partial{\mathrm{X}}$ .

We are going to construct a free subgroup of infinite rank with nontrivial conservative and dissipative components. The construction is motivated by examples in free groups given in [35].

11.1. Preparation

Pick up any non-pinched contracting element $f\in\Gamma$ and form the contracting system $\mathscr{F}=\{g\mathrm{Ax}(f):g\in\Gamma\}$ , whose stabilizers are accordingly $gE(f)g^{-1}$ .

Projection complex

For a sufficiently large $K>0$ , the projection complex $\mathcal{P}_{K}(\mathscr{F})$ defined by Bestvina-Bromberg-Fujiwara [9] is a quasi-tree of infinite diameter ([9]):

•

The vertex set consists of all elements in $\mathscr{F}$ .
•

Two vertices $u\neq v\in\mathscr{F}$ are connected by one edge if for any $w\in\mathscr{F}$ with $u\neq w\neq v$ , we have $\textbf{d}_{w}(u,v)>K$ . Here the projection is understood between $w,u,v\in\mathscr{F}$ as axes in $\mathrm{X}$ .

Remark 11.1.

Actually, the function “ $\textbf{d}_{w}$ ” is a slight modification of $\textbf{d}_{w}(u,v)=\|\pi_{w}(u)\cup\pi_{w}(v)\|$ , up to an additive amount. See [9, Def. 3.1] and [10, Sec. 4] for relevant discussions. We shall use the later one, which does not affect the estimates in practice.

Any two vertices $u\neq v\in\mathscr{F}$ are connected by a standard path $\gamma$ consisting of consecutive vertices

w_{0}:=u,w_{1},\cdots,w_{n}:=v

where $\textbf{d}_{w_{i}}(u,v)>K$ . See [9, Theorem 3.3]. This does not certifies the connectedness of $\mathcal{P}_{K}(\mathscr{F})$ , but also plays a crucial role in the whole theory. Among the useful facts, we mention

(1)

Standard paths are $2$ –quasi-geodesics ([10, Corollary 3.7]).
(2)

A triangle formed by three standard paths $\alpha,\beta,\gamma$ is almost a tripod: $\gamma$ is contained in the union $\alpha\cup\beta$ , with an exception of at most two consecutive vertices ([10, Lemma 3.6]).

By the rotating family theory, Dahmani-Guirardel-Osin [24] proved that

Theorem 11.2.

There exists a (not possibly every) large $n>0$ so that the normal closure $G:=\llangle f^{n}\rrangle$ is a free group of infinite rank.

We recall some ingredients in the proof using projection complex recently given in [11] inspired by Clay-Mangahas-Margalit [17].

Fix a basepoint $v_{0}:=\mathrm{Ax}(f)$ at $\mathcal{P}_{K}(\mathscr{F})$ , so $v_{0}$ is fixed by $E(f)<\Gamma$ . By definition, $G$ is generated by all conjugates of $\langle f^{n}\rangle$ in $\Gamma$ . If $n$ is chosen so $\langle f^{n}\rangle$ is normal in $E(f)$ , then $G$ acts on $\mathcal{P}_{K}(\mathscr{F})$ with the stabilizers exactly from those conjugates of $\langle f^{n}\rangle$ . In particular, the stabilizer of $v_{0}$ is now $\langle f^{n}\rangle$ .

If such $n\gg 0$ is large enough, the action $G\curvearrowright\mathcal{P}_{K}(\mathscr{F})$ is proved in [11, Thm 3.2] to be an $L$ –spinning family in sense of [17]:

(37)

\textbf{d}_{v_{0}}(w,f^{n}w)>L,\;\forall v_{0}\neq w\in\mathscr{F}

We follow closely the account of [11] to explain.

Construction of free base

We do it inductively and start with $G_{0}=\langle f^{n}\rangle$ and $S_{0}=\{v_{0}\}$ . Let $N_{1}$ be the $1$ –neighborhood of $W_{0}:=S_{0}$ and $S_{1}$ be the set of vertices up to $G_{0}$ –conjugacy in $N_{1}\setminus W_{0}$ . Set $G_{1}=\langle G_{v}:v\in S_{1}\cup S_{0}\rangle$ and $W_{1}=G_{1}N_{1}$ . Then $S_{0}\cup S_{1}$ is a fundamental set for the action $G_{1}\curvearrowright W_{1}^{0}$ , i.e. containing exactly one point from each orbit. Here $W_{1}^{0}$ denotes the vertex set of $W_{1}$ .

For $i\geq 2$ , let $W_{i}=G_{i}\cdot N_{i-1}$ and define $N_{i+1}$ to be the $1$ –neighborhood of $W_{i}$ . Let $S_{i+1}$ be the set of vertices up to $G_{i}$ –conjugacy in $N_{i+1}\setminus W_{i}$ , so that each vertex in $S_{i+1}$ is adjacent to some in $S_{i}$ . This can be done, as $W_{i}^{0}=G_{i}\cdot(\cup_{0\leq j<i}S_{j})$ . Then $G_{i+1}:=\langle G_{v}:v\in\cup_{0\leq j\leq i}S_{j}\rangle$ is generated by the stabilizers of vertices in $N_{i+1}$ .

Canoeing path

Recall that $K$ is a large constant chosen for $\mathcal{P}_{K}(\mathscr{F})$ . Any two vertices $u,v$ in $W_{i}$ can be connected by an $L$ –canoeing path $\gamma$ for some $L=L(K)>0$ (see [11, Def. 4.3 & Lemma 4.9]):

(1)

$\gamma$ is a concatenation $\gamma_{1}\cdot\gamma_{2}\cdot\cdots\cdot\gamma_{n}$ where $\gamma_{i}$ is a geodesic or the union of two geodesics.
(2)

The common endpoint $v$ of $\gamma_{i}$ and $\gamma_{i+1}$ has $L$ –large angle: if $v_{-}\in\gamma_{i},v_{+}\in\gamma_{i+1}$ are previous and next vertices, then $\textbf{d}_{v}(v_{-},v_{+})>L$ , where $v,v_{-},v_{+}\in\mathscr{F}$ are understood as axes in $\mathrm{X}$ .

By [11, Prop. 4.4], $L$ –large angle points are contained in the standard path from $u$ to $v$ .

In [11], through the canoe path, it is inductively proven that $G_{i+1}$ is generated by $G_{i}$ with a free group generated by $S_{i}$ . Exhausting $\mathcal{P}_{K}(\mathscr{F})$ by $W_{i}$ , $G$ is obtained as the direct limit of $G_{i}$ , which is a free group generated by stabilizers of vertices in $S:=\cup_{0\leq j<\infty}S_{j}$ . Recall that $G_{0}=\langle f^{n}\rangle$ and $S_{0}=\{v_{0}\}$ , and consider the splitting

G=G_{0}\star H

where $H:=\langle G_{v}:v\in S\setminus S_{0}\rangle$ is a free group. We also need the associated Bass-Serre tree $T$ . More precisely, as the quotient graph $T/G$ is an interval, we partition $T^{0}=V_{1}\cup V_{2}$ with the stabilizers of vertices in $V_{1}$ being conjugate to $G_{0}$ and the stabilizers of vertices in $V_{2}$ to $H$ . Moreover,

•

the set of $G_{0}$ –vertices (resp. $H$ –vertices) in $V_{1}$ (resp. $V_{2}$ ) are labeled by left $G_{0}$ –cosets (resp. $H$ –cosets) in $G$ ;
•

the set of edges adjacent to $u\in V_{1}$ (resp. $u\in V_{2}$ ) are bijective to the elements in $G_{0}$ (resp. $H$ ).

11.2. Subgroups with nontrivial conservative part

We now state the main theorem in this section.

Theorem 11.3.

The subgroup $H$ has proper limit set $[\Lambda(Ho)]\subsetneq[\Lambda(\Gamma o)]$ . Moreover, If $\omega_{G}<\omega_{\Gamma}$ , then $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})\geq\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})>0$ .

In particular, since $H$ has a proper limit set, it has maximal quotient growth by [36]. By Corollary 1.20, which will be proven in Section 10, Theorem 11.3 implies that the action of $H$ has non-trivial Hopf decomposition.

Remark 11.4.

In the construction, we have that $\Gamma/G$ is non-elementary acylindrically hyperbolic, in particular it is non-amenable. The amenability conjecture states that for a normal subgroup $G$ , $\omega_{G}=\omega_{\Gamma}$ if and only if the quotient group $\Gamma/G$ is amenable. If the amenability conjecture is verified for any SCC action, then the assumption that $\omega_{G}<\omega_{\Gamma}$ could be removed.

The remainder of this subsection is devoted to the proof of Theorem 11.3, which relies on the following decomposition of the limit set of $G$ .

The end boundary $\partial T$ of the Bass-Serre tree $T$ associated to $G_{0}\star H$ consists of all geodesic rays from a fixed point. As $T$ is locally infinite, it is non-compact. Note that $[\Lambda(G_{0}o)]=[f^{-}]\cup[f^{+}]$ for $G_{0}=\langle f^{n}\rangle$ . In what follows, we understand $[\Lambda(\Gamma o)]$ as the quotient space by identifying each $[\cdot]$ –class as one point. Recall that $\mathcal{C}$ is the set of non-pinched points in $\partial{\mathrm{X}}$ (Definition 2.13(C)), and the points outside $\mathcal{C}$ are pinched.

Proposition 11.5.

We have the following covering

(38)

[\Lambda(Go)]=\left[\bigcup_{g\in G}g[\Lambda(Ho)]\right]\bigcup\left[\bigcup_{g\in G}g[f^{\pm}]\right]\bigcup\iota(\partial T)

where a map $\iota:\partial T\to[\Lambda(\Gamma o)]$ is defined below in Lemma 11.7. Moreover, this covering becomes a disjoint union if every term on RHS and LHS is replaced by its intersection with the set of non-pinched points $\mathcal{C}$ .

In a variety of examples, it is known that $\partial{\mathrm{X}}=\mathcal{C}$ consists of non-pinched points, so the union in (38) becomes a disjoint union. We record this for further reference.

Corollary 11.6.

Suppose that $(\mathrm{X},\partial{\mathrm{X}})$ is one of the items (1),(2),(3) and (5) in Examples 2.15. Then the following holds:

(39)

[\Lambda(Go)]=\left[\bigsqcup_{g\in G}g[\Lambda(Ho)]\right]\bigsqcup\left[\bigsqcup_{g\in G}g[f^{\pm}]\right]\bigcup\iota(\partial T)

where a map $\iota:\partial T\to[\Lambda(\Gamma o)]$ is defined below in Lemma 11.7.

As shown in (39), the first points of any contracting elements other than $f$ must be contained in some $G$ –translate of $\Lambda(Ho)$ .

In the next three steps, we shall describe in detail how a geodesic on the Bass-Serre tree $T$ produces a canoeing path in the projection complex $\mathcal{P}_{K}(\mathscr{F})$ , which is then lifted to an admissible path in the space $\mathrm{X}$ . This is the crux of the proof of Proposition 11.5.

Let $U$ be a non-empty reduced word over the alphabet $(H\cup G_{0})\setminus 1$ . We decompose $U$ into nonempty subwords $U_{i}$ separated by letters in $G_{0}=\langle f^{n}\rangle$ :

(40)

\displaystyle U=(h_{1},f^{n_{1}},h_{2},f^{n_{2}},\cdots,h_{k},f^{n_{k}})

where $h_{i}\in H$ and $n|n_{i}\neq 0$ unless the last one $i=k$ . For each $h_{i}$ , we have a minimal $j=j(U_{i})$ so that $h_{i}$ represents an element in $G_{j}\setminus G_{0}\subseteq H$ . Then the element $g=h_{1}f^{n_{1}}\cdots h_{k}f^{n_{k}}$ is represented by $U$ . If $g$ is not in $H$ , then $k\geq 2$ and $n_{1}\neq 0$ .

Fix an oriented edge $e=[\mathbf{v}_{0},\mathbf{v}_{1}]$ in $T$ with endpoint stabilizers $G_{0}$ and $H$ respectively. Denote $\overleftarrow{e}=[\mathbf{v}_{1},\mathbf{v}_{0}]$ the reversed edge. We consider the basepoint $\mathbf{v}_{0}$ corresponding to the coset $G_{0}$ in $T$ , and the basepoint $v_{0}=\mathrm{Ax}(f)$ in $\mathcal{P}_{K}(\mathscr{F})$ stabilized by $G_{0}$ .

Figure 11. Unfolding the word

U

as a geodesic

\alpha

in Bass-Serre tree

T

. The reader should be cautioned that the labels indicate the action of stabilizers on vertices understood up to appropriate conjugation.

Step 1

The word $U$ gets unfolded as a geodesic $\alpha$ in $T$ starting with the edge $e$ , at which $h_{1}\in H$ rotates the edge $e$ at $\mathbf{v}_{1}$ to $h_{1}e$ . Rotating edges consecutively in this way, we write down

\alpha=e\cdot(h_{1}\overleftarrow{e})\cdot(h_{1}f^{n_{1}}e)\cdot(h_{1}f^{n_{1}}h_{2}\overleftarrow{e})\cdot\cdots\cdot(h_{1}f^{n_{1}}\cdots h_{k}\overleftarrow{e})

where the last edge terminates at the endpoint fixed by corresponding conjugate of $f^{n_{k}}$ . See Fig. 11.

Step 2

Recall $W_{j(i)}=G_{j(i)}\cdot N_{j(i)-1}$ and $h_{i}\in G_{j(i)}$ . As mentioned above, there exists an $L$ –canoeing path $\gamma_{i}$ in $W_{j(i)}$ from $v_{0}$ to $h_{i}v_{0}$ . These $h_{i}^{-1}\gamma_{i}$ and $\gamma_{i+1}$ are adjacent at $v_{0}$ , around which $f^{n_{i}}$ rotates $\gamma_{i+1}$ . By (37), $d_{v_{0}}(v,f^{n_{i}}v)>L$ for any $v\neq v_{0}$ , so $h_{i}v_{0}$ is a $L$ –large angle point of the concatenation $\gamma_{i}\cdot(h_{i}f^{n_{i}}\gamma_{i+1})$ . We then obtain an $L$ –canoeing path $\gamma$ from $v_{0}$ to $gv_{0}$ as the concatenation of $\gamma_{i}$ $(1\leq i\leq n)$ up to appropriate translation,

\gamma=\gamma_{1}\cdot(h_{1}f^{n_{1}}\gamma_{2})\cdot(h_{1}f^{n_{1}}h_{2}f^{n_{2}}\gamma_{3})\cdot\cdots\cdot(h_{1}f^{n_{1}}\cdots h_{k-1}f^{n_{k-1}}\gamma_{k})

where the common endpoints of two consecutive paths are $L$ –large angle points fixed by the corresponding conjugates of $f^{n_{i}}$ . See Fig. 12.

Figure 12. Canoeing path

\gamma

\mathcal{P}_{K}(\mathscr{F})

: join canoeing paths

\gamma_{i}

at large angle points

h_{1}f^{n_{1}}\cdots h_{i}f^{n_{i}}v_{0}

. Those large points are contained in the standard path

\beta

(possibly

\beta\neq\gamma

) from

v_{0}

gv_{0}

Step 3

By [11, Prop. 4.4], those $L$ –large angle points lie on the standard path $\beta$ from $v_{0}$ to $gv_{0}$ . We lift the standard path $\beta$ to get an $(\hat{L},\tau)$ –admissible path $\tilde{\beta}$ in $\mathrm{X}$ as follows, with contracting sets given by the axis $w_{i}:=h_{1}f^{n_{1}}\cdots h_{i}\mathrm{Ax}(f)$ where $w_{0}=\mathrm{Ax}(f),w_{k+1}=g\mathrm{Ax}(f)$ :

Choose two points $o\in w_{0}$ and $go\in w_{k+1}$ in $\mathrm{X}$ . Connect $o,go$ by concatenating geodesics in consecutive axis $w_{i}$ and then geodesics from $\pi_{w_{i-1}}(w_{i})$ to $\pi_{w_{i}}(w_{i-1})$ . See a schematic illustration (13) of a lifted path.

By [72, Lemma 2.22], this gives an $(\hat{L},\tau)$ –admissible path from $o$ to $go$ with associated contracting axes $w_{i}$ ’s. The constant $\hat{L}$ is comparable with $L$ and $\tau$ depends only on $\mathscr{F}$ . See [72] for details.

Figure 13. Lift a standard path

\beta

(at bottom) in

\mathcal{P}_{K}(\mathscr{F})

to admissible path (at top) in

\mathrm{X}

We first establish the desired map mentioned in Proposition 11.5.

Lemma 11.7.

There exists a map $\iota:\partial T\to[\Lambda(\Gamma o)]$ so that the image $\iota(\partial T)$ is contained in the conical limit set $[\Lambda^{\mathrm{con}}(Go)]$ . Moreover, if $\iota(p)=\iota(q)$ for $p\neq q$ , then $\iota(p)$ is outside the set $\mathcal{C}$ of non-pinched points.

Proof.

By construction, $T^{0}=\{gH,gG_{0}:g\in G\}$ , so any geodesic ray $\alpha$ in $T$ from $v_{0}=G_{0}$ has the list of all vertices $A_{n}$ , which are alternating left $H$ and $G_{0}$ cosets. This determines a unique infinite word of form (40). By Step (3), the lifted $(\tilde{L},\tau)$ –admissible path $\tilde{\beta}$ in $\mathrm{X}$ intersects infinitely many translates $X_{n}$ of $N_{r}(\mathrm{Ax}(f))$ under $G$ , as $A_{n}$ contains infinitely many $G_{0}$ –cosets. By Definition 2.13(B), $X_{n}$ accumulates into a $[\cdot]$ –class $[\xi]$ . If we choose a sequence $g_{n}\in A_{n}$ , we see $d(g_{n}o,[o,\xi])\leq r$ so $\xi$ lies in $[\Lambda^{\mathrm{con}}(Go)]$ . Setting $\iota(\alpha^{+})=[\xi]$ defines the desired map $\iota:\partial T\to[\Lambda^{\mathrm{con}}(Go)]$ .

To justify the claim of injectivity, consider a distinct geodesic $\alpha\neq\beta$ originating at $v_{0}$ with the set of vertices $B_{n}$ , so that $[\xi]=\iota(\alpha^{+})=\iota(\beta^{+})$ . We shall show that $[\xi]$ is pinched. Let $\gamma$ be the bi-infinite geodesic in the tree $T$ from $\alpha^{+}$ to $\beta^{+}$ , i.e.: $\gamma=(\alpha\cup\beta)\setminus[v_{0},w_{0})$ as a set, where $w_{0}$ is the vertex that $\beta$ departs from $\alpha$ . As shown in the first paragraph, there are two sequences $g_{n}o\in A_{n}o$ and $g^{\prime}_{n}o\in B_{n}o$ tending to $[\xi]$ .

We read off from $\gamma$ a bi-infinite alternating word $U$ in the form (40). As described in Step (3), we obtain from $\gamma_{n}$ an $(\tilde{L},\tau)$ –admissible path $\hat{\gamma}_{n}$ from $g_{n}o$ to $g_{n}^{\prime}o$ , where the contracting subsets correspond to the $G_{0}$ –vertices on $\gamma$ . By Proposition 2.9, $[g_{n},g_{n}^{\prime}o]$ has at most $r$ –distance to the entry and exit points of the contracting subsets. See Fig. 3. Recall that $T$ is a bipartite graph with $G_{0}$ –vertices and $H$ –vertices. Look at the first $G_{0}$ –vertex $w$ on $\beta$ after $w_{0}$ , and its exit point $x$ of the contracting subset. By the above discussion, we have $d(x,[g_{n}o,g_{n}^{\prime}o])\leq r$ , that is $[g_{n}o,g^{\prime}_{n}o]$ is non-escaping. Hence, $[\xi]\notin\mathcal{C}$ is pinched. ∎

We continue to examine the possible intersection in (38).

Lemma 11.8.

The following hold.

(1)

For any $g\in G\setminus H$ , the intersection $[\Lambda(Ho)]\cap g[\Lambda(Ho)]$ consist of only pinched points.
(2)

$[\Lambda(Ho)]$ intersects $\iota(\partial T)$ only in pinched points.
(3)

$[\Lambda(Ho)]\cap g[f^{\pm}]=\emptyset$ for any $1\neq g\in G$ .

Proof.

(1). Without loss of generality, we may represent $g$ as a nonempty alternating word $U=\mathfrak{h}_{1}f^{n_{1}}\cdots\mathfrak{h}_{k}f^{n_{k}}$ as in (40), ending with letters in $G_{0}$ . Let $h_{n}o\in Ho$ and $g_{n}o\in gHo$ tend to the same $[\xi]$ . We will show $[\xi]\notin\mathcal{C}$ ia pinched: that is, $\gamma_{n}:=[h_{n}o,g_{n}o]$ intersects a compact set for all $n\geq 1$ .

To this end, consider the nonempty alternating word $W$ representing $h_{n}^{-1}g_{n}$ . As $h_{n}\in H$ and $g_{n}\in gH$ , the word $W$ has the form $\tilde{h}_{n}f^{n_{1}}\cdots\mathfrak{h}_{k}f^{n_{k}}$ , where $\tilde{h}_{n}=h_{n}^{-1}\mathfrak{h}_{1}$ gives one letter in $H$ . We obtain from $W$ , as described in the Step (3) above, an $(L,\tau)$ –admissible path $\beta$ , which has the same endpoints $o,h_{n}^{-1}g_{n}o$ as $h_{n}^{-1}\gamma_{n}$ . Note that $\mathfrak{h}_{1}f^{n_{1}}o$ is the exit vertex of the admissible path $h_{n}\beta$ . Up to translation, we obtain that $\gamma_{n}$ has $r$ –distance to the fixed point $\mathfrak{h}_{1}f^{n_{1}}o$ by Proposition 2.9. This shows that $[\xi]$ lies outside of the non-pinched points $\mathcal{C}$ .

(2). The same reasoning shows that $\Lambda(Ho)$ intersects $\iota(\partial T)$ only in pinched points. Indeed, if $[\xi]$ lies in $\iota(\partial T)$ , then it is an accumulation point of $g_{n}o$ , where $g_{n}$ lies in infinitely many $G_{0}$ –cosets (see the proof of Lemma 11.7). The same argument as above implies that $\gamma_{n}$ is non-escaping.

(2). The same argument proves that $\Lambda(Ho)\cap g[f^{\pm}]$ must be pinched. However, since $[f^{\pm}]$ is assumed to be non-pinched points, we obtain $\Lambda(Ho)\cap g[f^{\pm}]=\emptyset$ . ∎

It is now fairly easy to derive from Lemma 11.7 and Lemma 11.8:

Proof of Proposition 11.5.

Note that $G$ is the union of all left $G_{0}$ –cosets and $H$ –cosets. Consider a limit point $\xi\in[\Lambda(Go)]$ in the following complement

[\Lambda(Go)]\setminus\left[\bigcup_{g\in G}g[\Lambda(Ho)]\right]\bigcup\left[\bigcup_{g\in G}g[\Lambda(G_{0}o)]\right]

so that $g_{n}o\to[\xi]$ , for a sequence of $g_{n}\in G$ belonging to infinitely many distinct $G_{0}$ –cosets and/or $H$ –cosets denoted by $A_{n}$ . By the above construction of Bass-Serre tree, $A_{n}$ corresponds to vertices on $T$ , so after passing to subsequence, we obtain a limiting geodesic ray $\alpha$ in $T$ starting at $\mathbf{v_{0}}$ : that is, the intersection of $\alpha$ with $[\mathbf{v_{0}},g_{n}\mathbf{v_{0}}]$ becomes unbounded as $n\to\infty$ . By the proof of Lemma 11.7, this implies $\xi$ is contained in $\iota(\partial T)$ .

The “moreover” statement follows from Lemma 11.8. The proof is now complete. ∎

Proof of Theorem 11.3.

First of all, as $G<\Gamma$ is an infinite normal subgroup, $\mu_{o}(\Lambda^{\mathrm{Hor}}(Go))=\mu_{o}(\Lambda^{\mathrm{hor}}(Go))=1$ by Theorem 1.11.

By Proposition 11.5, $[\Lambda(Go)]$ is covered by a countable union $G\cdot[\Lambda(Ho)]$ with the remaining “exceptional” set $E$ :

E:=\left[\bigcup_{g\in G}g[\Lambda(G_{0}o)]\right]\cup\iota(\partial T).

As $\iota(\partial T)$ is contained in the conical points, so is shadowed by the elements in $Go$ . By the assumption that $\omega_{G}<\omega_{\Gamma}$ , we have $\iota(\partial T)$ is $\mu_{o}$ –null by Lemma 2.34. This then proves $\mu_{o}(E)=0$ .

This concludes the proof modulo of the following claim.

Claim.

A small/big horospheric limit points of $G$ is either in $E$ or a small/big horospheric limit point in a $G$ –translate of $\Lambda(Ho)$ . That is,

[\Lambda^{\mathrm{Hor}}(Go)]\subseteq E\bigcup\left[\bigcup_{g\in G}g[\Lambda^{\mathrm{Hor}}(Ho)]\right]

Proof of the claim.

Indeed, let $\xi\in\Lambda^{\mathrm{Hor}}(Go)\setminus E$ . Then there exists a sequence $g_{n}o\in Go$ tending to $[\xi]$ in some (or any) horoball $\mathcal{HB}([\xi])$ based at $[\xi]$ . As $\xi\notin E$ , $g_{n}\in G$ lies in finitely many $H$ –cosets, and after passing to subsequence, we may assume $g_{n}H=g_{0}H$ for $n\gg 1$ . By Lemma 11.8, $g_{0}\Lambda(Ho)\cap\Lambda(Ho)=\emptyset$ for $g_{0}\notin H$ , so we conclude that $g_{0}H=H$ and thus $\xi\in\Lambda^{\mathrm{Hor}}(Ho)$ . ∎

As $E$ is $\mu_{o}$ –null, we thus obtain from Proposition 11.5 that $\mu_{o}(\Lambda^{\mathrm{Hor}}{(Ho)})\geq\mu_{o}(\Lambda^{\mathrm{hor}}{(Ho)})>0$ . ∎

We conclude this section with a corollary which might be of independent interest. Define a topology on $M:=T^{0}\cup\partial T$ as follows (cf. [23]). A sequence of points $x_{n}\to x$ if and only if the geodesic $[x_{n},x]$ eventually passes no element in any given finite set of edges (which is required adjacent to $x$ if $x\in T^{0}$ ). This makes $M$ homeomorphic to a Cantor space, in particular, it is compact. If $T$ is the Bass-Serre tree of $\Gamma=H\star K$ , then $M$ is exactly the Bowditch boundary of a hyperbolic group $\Gamma$ relative to $\{H,K\}$ . Moreover, if both $H,K$ are one-ended, it is also the end boundary of $\Gamma$ .

Corollary 11.9.

In the setting of Corollary 11.6, there exists a continuous surjective $G$ –map from $\Lambda(Go)$ to $M$ , where $M$ can be identified with the Bowditch boundary of $G$ relative to factors $G_{0}$ and $H$ .

The reader may have noticed that item (4) in Examples 2.15 is not covered in the corollaries. It is not clear whether the decomposition holds on the Thurston boundary without taking the intersection with $\mathscr{UE}$ .

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Confined subgroups in groups with contracting elements

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

Definition 1.1.

Examples 1.2.

1.1. Main applications

Theorem 1.3.

Theorem 1.4.

Remark 1.5.

Theorem 1.6.

Remark 1.7.

1.2. Ergodic properties of boundary actions

Theorem 1.8 (HTS dichotomy for CAT(-1) spaces).

Examples 1.9.

Theorem 1.10.

Theorem 1.11.

Corollary 1.12.

1.3. Growth inequalities for confined subgroups

Theorem 1.13.

Corollary 1.14.

Theorem 1.15.

Corollary 1.16.

Proofs of Theorems 1.3, 1.4 and 1.6.

1.4. Maximal quotient growth

Theorem 1.17.

Question 1.18.

Theorem 1.19.

Corollary 1.20.

Theorem 1.21.

1.5. Ingredients in the proofs and organization of the paper

Results on Hopf decomposition

Key technical results on confined subgroups

Lemma 1.22 (=Lemma 5.8).

Lemma 1.23 (Shadow Principle in §7).

Cogrowth tightness of confined subgroups

Subgroups with nontrivial Hopf decomposition

A guide to the sections of the paper

2. Preliminaries

2.1. Contracting geodesics

Definition 2.1.

Lemma 2.2.

Definition 2.3.

Remark 2.4.

Definition 2.5.

2.2. Extension Lemma

Definition 2.6 (Admissible Path).

Remark 2.7.

Definition 2.8 (Fellow travel).

Proposition 2.9.

Lemma 2.10 (Extension Lemma).

Remark 2.11.

Lemma 2.12.

2.3. Horofunction boundary

Finite difference relation.

2.4. Convergence boundary

Definition 2.13.

Remark 2.14.

Examples 2.15.

Theorem 2.16.

Proof.

Lemma 2.17.

Proof.

Lemma 2.18.

Lemma 2.19.

Lemma 2.20.

Proof.

Lemma 2.21.

2.5. Quasi-conformal density and Patterson’s construction

Assumption D.

Horofunctions in convergence boundary

Horoballs at [⋅][\cdot]–classes

Definition 2.22.

Lemma 2.23.

Proof.

Definition 2.24.

Remark 2.25.

Patterson’s construction of quasi-conformal density

Example 2.26.

Horoballs at $[\cdot]$ –classes

Lemma 3.4 ( $\ll$ Lemma 5.7).

Lemma 3.5 ( $\ll$ Lemma 5.8).