An extended definition of Anosov representation for relatively hyperbolic groups

This paper constitutes the first part of a preprint originally posted under the same title in April 2022, and covers the definition of an EGF representation and the proof of the main stability theorem. The only changes to the contents are minor corrections and some reworked exposition. For the sake of shortening the paper, the second part of the earlier preprint (covering examples of EGF representations) will be posted as a separate article [Wei23b]. The contents of the sequel are also largely unchanged from their form in the original preprint.

Anosov representations are a family of discrete representations of hyperbolic groups into reductive Lie groups, and give a natural higher-rank generalization of convex cocompact representations in rank-one. They were originally defined for surface groups by Labourie [Lab06], and the definition was extended to general word-hyperbolic groups by Guichard-Wienhard [GW12].

Many equivalent characterizations of Anosov representations have since been uncovered [KLP14, KLP17, GGKW17, BPS19, Tso20, KP22], and much of the theory of convex cocompact groups in rank-one has been extended to this higher-rank context. For instance, Anosov representations satisfy a stability property: if $\Gamma$ is a hyperbolic group, then the Anosov representations from $\Gamma$ into a semisimple Lie group $G$ form an open subset of the representation variety $\operatorname{Hom}(\Gamma,G)$.

The purpose of this paper is to introduce a higher-rank notion of geometrical finiteness, which involves generalizing the notion of an Anosov representation to allow for a relatively hyperbolic domain group. Our definition—that of an extended geometrically finite (EGF) representation—encompasses all previous definitions of relative Anosov representation, and additionally covers various forms of higher-rank “geometrical finite” behavior which are not described by other definitions.

Our starting point is a definition of Anosov representation in terms of topological dynamics: if $\Gamma$ is a hyperbolic group, $G$ is a semisimple Lie group, and $P\subset G$ is a symmetric parabolic subgroup, a representation $\rho:\Gamma\to G$ is $P$-Anosov if there is a $\rho$-equivariant embedding $\xi:\partial\Gamma\to G/P$ of the Gromov boundary $\partial\Gamma$ of $\Gamma$ satisfying certain dynamical properties.

Existing notions of relative Anosov representations (see e.g. [KL18], [Zhu21], [ZZ22]) replace $\xi$ with an embedding of the Bowditch boundary of a relatively hyperbolic group. This condition forces the limit set in $G/P$ of a peripheral subgroup to be a singleton, so that the peripheral subgroup itself must be both weakly unipotent and regular (see [KL18, Section 5]). These requirements are natural in rank one, but less so in higher rank, and there are a number of interesting examples where they are not satisfied (see Section 1.6).

The main idea behind the definition of an extended geometrically finite representation is to reverse the direction of the boundary map: we characterize geometrical finiteness via the existence of an equivariant map from a closed subset of a flag manifold to the Bowditch boundary of a relatively hyperbolic group, rather than the other way around. Our “backwards” boundary map does not need to be a homeomorphism, so the approach allows for more flexibility in the higher rank setting; in particular, there is no restriction on the limit set of a peripheral subgroup in $G/P$, or even a requirement that this “limit set” is well-defined. Moreover, even in rank one, this new perspective seems useful for studying deformations of discrete groups which do not preserve the homeomorphism type of the limit set (see Section 1.8.3).

With the additional flexibility afforded by the definition, we can prove a general stability result for EGF representations (see Section 1.4 below). It is not true that an arbitrary (sufficiently small) deformation of an EGF representation is still EGF; indeed, it is possible to find small deformations of geometrically finite representations in rank one which are not even discrete. However, we prove that any EGF representation $\rho:\Gamma\to G$ is relatively stable: any small deformation of $\rho$ in $\operatorname{Hom}(\Gamma,G)$ which satisfies a condition on the peripheral subgroups is also EGF. The peripheral condition—which we call peripheral stability—is general enough to hold even in the absence of a topological conjugacy between the limit sets of the original peripheral subgroups and their deformations. In specific cases, it can even be used to deduce absolute stability results for non-hyperbolic groups (see the last section of [Wei23b]).

If $\Gamma$ is a relatively hyperbolic group, relative to a collection $\mathcal{H}$ of peripheral subgroups, then $\Gamma$ acts as a convergence group on the Bowditch boundary $\partial(\Gamma,\mathcal{H})$. We recall the definition here.

When $\gamma_{n}$ is a sequence of distinct elements in a relatively hyperbolic group $\Gamma$, then $\gamma_{n}$ converges to $b$ uniformly on compacts in $\partial(\Gamma,\mathcal{H})-\{a\}$ if and only if $\gamma_{n}$ converges to $b$ in the compactification $\overline{\Gamma}=\Gamma\sqcup\partial(\Gamma,\mathcal{H})$ and the inverse sequence $\gamma_{n}^{-1}$ converges to $a$.

Recall that if a group $\Gamma$ acts by homeomorphisms on a Hausdorff space $X$, the pair $(\Gamma,X)$ is called a topological dynamical system. We say that an extension of $(\Gamma,X)$ is a topological dynamical system $(\Gamma,Y)$ together with a $\Gamma$-equivariant surjective map $\phi:Y\to X$.

In this paper, when $(\Gamma,\mathcal{H})$ is a relatively hyperbolic pair (i.e. $\Gamma$ is hyperbolic relative to a collection $\mathcal{H}$ of peripheral subgroups), we will consider embedded extensions of the topological dynamical system $(\Gamma,\partial(\Gamma,\mathcal{H}))$. We want these embedded extensions to respect the convergence group action of $(\Gamma,\partial(\Gamma,\mathcal{H}))$ in some sense, so we introduce the following definition: {restatable}definitiondynamicsPreserving Let $(\Gamma,\mathcal{H})$ be a relatively hyperbolic pair, with $\Gamma$ acting on a connected compact metrizable space $M$ by homeomorphisms. Let $\Lambda\subset M$ be a closed $\Gamma$-invariant set.

We say that a continuous equivariant surjective map $\phi:\Lambda\to\partial(\Gamma,\mathcal{H})$ extends the convergence action of $\Gamma$ if for each $z\in\partial(\Gamma,\mathcal{H})$, there exists an open set $C_{z}\subset M$ containing $\Lambda-\phi^{-1}(z)$, satisfying the following:

If $\gamma_{n}$ is a sequence in $\Gamma$ with $\gamma_{n}^{\pm 1}\to z_{\pm}$ for $z_{\pm}\in\partial(\Gamma,\mathcal{H})$, then for any compact set $K\subset C_{z_{-}}$ and any open set $U$ containing $\phi^{-1}(z_{+})$, for sufficiently large $n$, $\gamma_{n}\cdot K$ lies in $U$.

Now let $G$ denote a semisimple Lie group with no compact factor. The central definition of the paper is the following: {restatable}definitionEGF Let $\Gamma$ be a relatively hyperbolic group, let $\rho:\Gamma\to G$ be a representation, and let $P\subset G$ be a symmetric parabolic subgroup. We say that $\rho$ is extended geometrically finite (EGF) with respect to $P$ if there exists a closed $\rho(\Gamma)$-invariant set $\Lambda\subset G/P$ and a continuous $\rho$-equivariant surjective antipodal map $\phi:\Lambda\to\partial(\Gamma,\mathcal{H})$ extending the convergence action of $\Gamma$.

The map $\phi$ is called a boundary extension of the representation $\rho$, and the closed invariant set $\Lambda$ is called the boundary set.

We refer to Section 4 for the definition of “antipodal map” in this context.

Like (relative) Anosov representations, extended geometrically finite representations are always discrete with finite kernel (see 4.1). The central result of this paper says that EGF representations have a relative stability property: if $\rho$ is an EGF representation, then certain small relative deformations of $\rho$ must also be EGF.

To state the theorem, we define a notion of a peripherally stable subspace of $\operatorname{Hom}(\Gamma,G)$. The precise definition is given in Section 9, but roughly speaking, a subspace $\mathcal{W}\subseteq\operatorname{Hom}(\Gamma,G)$ is peripherally stable if the large-scale dynamical behavior of the peripheral subgroups of $\Gamma$ is in some sense preserved by small deformations inside of $\mathcal{W}$. We emphasize again that the action of a deformed peripheral subgroup does not need to be even topologically conjugate to the action of the original peripheral subgroup.

We prove the following: {restatable}theoremcuspStableStability Let $\rho:\Gamma\to G$ be EGF with respect to $P$, let $\phi:\Lambda\to\partial(\Gamma,\mathcal{H})$ be a boundary extension, and let $\mathcal{W}\subseteq\operatorname{Hom}(\Gamma,G)$ be peripherally stable at $(\rho,\phi)$. For any compact subset $Z$ of $\partial(\Gamma,\mathcal{H})$ and any open set $V\subset G/P$ containing $\phi^{-1}(Z)$, there is an open subset $\mathcal{W}^{\prime}\subset\mathcal{W}$ containing $\rho$ such that each $\rho^{\prime}\in\mathcal{W}^{\prime}$ is EGF with respect to $P$, and has an EGF boundary extension $\phi^{\prime}$ satisfying $\phi^{\prime-1}(Z)\subset V$.

While the peripheral stability condition in Section 1.4 is mildly technical, we can also apply it to yield more concrete results.

1.4 gives a very restrictive example of a peripherally stable subspace of $\operatorname{Hom}(\Gamma,G)$. But, in general the peripheral stability condition is flexible enough to allow peripheral subgroups to deform in nontrivial ways.

In particular, it is possible to find peripherally stable deformations of an EGF representation $\rho:\Gamma\to\operatorname{PGL}(d,\mathbb{R})$ which change the Jordan block decomposition of elements in the peripheral subgroups. For instance, one can deform an EGF representation in $\operatorname{PGL}(d,\mathbb{R})$ with unipotent peripheral subgroups into an EGF representation with diagonalizable peripheral subgroups—see Example 9.3.

Our proof of Section 1.4 is loosely inspired by Sullivan’s proof of stability for convex cocompact groups in rank-one [Sul85], but with substantial modification and several new ideas needed. Sullivan’s original approach was to show that a discrete group $\Gamma\subset\operatorname{PO}(d,1)$ is convex cocompact if and only if the action of $\Gamma$ on its limit set $\Lambda$ in $\partial\mathbb{H}^{d}$ satisfies an expansion property. Then, he used this expansion property to give a symbolic coding for infinite quasigeodesic rays in $\Gamma$. This coding gives a way to see that the correspondence between geodesic rays in $\Gamma$ and points in $\partial\mathbb{H}^{d}$ is stable under small perturbations of the representation.

Other authors have successfully used expansion dynamics and symbolic codings to prove stability results in the higher-rank setting (see [KKL19], [BPS19]). In our context, however, we encounter several new challenges. The first problem is that EGF representations do not actually have “expansive” or even “relatively expansive” dynamics on their limit sets in $G/P$ in any metric sense. This means we need a way to understand “relative expansion” (or its inverse, “relative contraction”) purely topologically. Second, we need to come up with a procedure for constructing a relative coding for points in the Bowditch boundary $\partial(\Gamma,\mathcal{H})$ of a relatively hyperbolic group $\Gamma$, in a way which is compatible with the “topological” expansion of the $\Gamma$-action on both $\partial(\Gamma,\mathcal{H})$ and $G/P$. In addition, we need to ensure that this compatibility is preserved under a wide variety of possible perturbations of the action of a peripheral subgroup.

The solution for all of these problems is to work almost entirely within the framework of extended convergence group actions. Using this setup, we provide a general construction for a relative quasi-geodesic automaton: a finite directed graph with edges labeled by subsets of $\Gamma$, so that paths in the graph are in rough correspondence with quasi-geodesics in the coned-off Cayley graph for the pair $(\Gamma,\mathcal{H})$. Each vertex $v$ of the graph is assigned an open subset $W_{v}$ of some compact metrizable space $M$ on which $\Gamma$ acts as an extended convergence group. Each edge $u\to v$ in the graph corresponds to a set of inclusions of the form

(1)

$$\alpha\cdot\overline{N_{\varepsilon}(W_{v})}\subset W_{u},$$

where $\alpha$ is an element in the subset of $\Gamma$ labeling the edge $u\to v$, and $N_{\varepsilon}(W_{v})$ is a small neighborhood of $W_{v}$ with respect to some fixed metric on $M$. This strong nesting of subsets turns out to be a reasonable stand-in for a “contraction” property of the group action. Each inclusion of the form (1) is an open condition in the $C^{0}$ topology on actions of $\Gamma$ on $M$, so when the set of elements labeling an edge $u\to v$ is finite, the corresponding set of inclusions are stable under small perturbations of the $\Gamma$-action. If the label of $u\to v$ is instead infinite, then the corresponding (infinite) set of inclusions may not be stable under arbitrary deformations—this is where the peripheral stability assumption is needed.

In its simplest form, our approach can be thought of as a “generalized ping-pong” argument: if $A,B$ are arbitrary finitely generated groups, then the group $A*B$ is hyperbolic relative to the collection of conjugates of $A$ and $B$. In this case, the inclusions in (1) are precisely the inclusions of sets required to set up a ping-pong argument proving that a representation of $A*B$ is discrete and faithful.

The related paper [Wei23b] (originally the second part of the current article) is focused on describing examples of EGF representations. Here, we briefly explain the nature of these examples, as well as additional examples described by other authors. See the references for further detail.

Previously, Kapovich-Leeb [KL18] and Zhu [Zhu21] independently introduced several notions of a relative Anosov representation. Later work of Zhu-Zimmer [ZZ22] showed that Zhu’s definition (that of a relatively dominated representation) is equivalent to one of the Kapovich-Leeb definitions (specifically, the definition of a relatively asymptotically embedded representation). In the special case where the domain group is isomorphic to a Fuchsian group, these definitions also agree with a notion of relative Anosov representation for Fuchsian groups introduced by Canary-Zhang-Zimmer [CZZ21].

Extended geometrically finite representations give a strict generalization of all of these definitions. We can precisely characterize when an EGF representation satisfies the stronger definition as well:

{restatable}

theoremrelAsympBdryDynamics Let $(\Gamma,\mathcal{H})$ be a relatively hyperbolic pair, and let $P\subset G$ be a symmetric parabolic subgroup. A representation $\rho:\Gamma\to G$ is relatively $P$-Anosov (in the sense given above) if and only if $\rho$ is EGF with respect to $P$, and has an injective boundary extension $\phi:\Lambda\to\partial(\Gamma,\mathcal{H})$.

We begin by providing some background in Sections 2 and 3, and then give the full formal definition of EGF representations in Section 4. In that section we also prove Section 1.7 (giving the connection between EGF representations and relative Anosov representations) and Theorem 1.11 (the Anosov relativization theorem). Some of these proofs assume the results of later sections, but they are not relied upon anywhere else in the paper.

The rest of the paper is devoted to the proof of our main stablity theorem for EGF representations (Section 1.4). In Section 5 and Section 6, we develop the main technical tool needed for the proof, which involves using the notion of an extended convergence group action to construct the relative quasigeodesic automaton alluded to previously. Then, in Section 7, we prove a key result (7.11) regarding a metric on certain open subsets of flag manifolds $G/P$, which we use to relate the results of the previous sections to relatively hyperbolic group actions on $G/P$. Then, we use all of these tools to develop an alternative characterization of EGF representations in Section 8, and finally prove our main theorem in Section 9.

The author thanks his PhD advisor, Jeff Danciger, for encouragement and many helpful conversations—without which this paper could not have been written. The author also thanks Katie Mann and Jason Manning for assistance simplifying some of the arguments in Sections 5 and 6. Further thanks are owed to Daniel Allcock, Dick Canary, Fanny Kassel, Max Riestenberg, Feng Zhu, and Andy Zimmer for providing feedback on various versions of this project.

This work was supported in part by NSF grants DMS-1937215 and DMS-2202770.

Recall that a finitely generated group $\Gamma$ is hyperbolic (or word-hyperbolic or $\delta$-hyperbolic or Gromov-hyperbolic) if and only if it acts properly discontinuously and cocompactly on a $\delta$-hyperbolic proper geodesic metric space $Y$.

A relatively hyperbolic group is also a group with an action by isometries on a $\delta$-hyperbolic proper geodesic metric space $Y$, but instead of asking for the action to cocompact, we ask for the action to be in some sense “geometrically finite.”

To be precise, this means that $Y$ has a $\Gamma$-invariant decomposition into a thick part $Y_{\mathrm{th}}$ and a countable collection $\mathcal{B}$ of horoballs. For a horoball $B$, we let $\mathrm{ctr}(B)$ denote the center of $B$ in $\partial Y$, and we let $\Gamma_{p}$ denote the stabilizer of any $p\in\partial Y$.

A group $\Gamma$ might be hyperbolic relative to different collections $\mathcal{H}$, $\mathcal{H}^{\prime}$ of peripheral subgroups. The collection $\mathcal{H}$ of peripheral subgroups is sometimes called a peripheral structure for $\Gamma$.

In general there is not a unique choice of Gromov model for a given relatively hyperbolic pair $(\Gamma,\mathcal{H})$, even up to quasi-isometry. There are various “canonical” constructions for a preferred quasi-isometry class of Gromov model, with certain desirable metric properties (see e.g. [Bow12], [GM08]).

Given any two Gromov models $Y$, $Y^{\prime}$ for $(\Gamma,\mathcal{H})$, there is always a $\Gamma$-equivariant homeomorphism $\partial Y\to\partial Y^{\prime}$ [Bow12]. The $\Gamma$-space $\partial Y$ is the Bowditch boundary of $(\Gamma,\mathcal{H})$. We will denote it by $\partial(\Gamma,\mathcal{H})$, or sometimes just $\partial\Gamma$ when the collection of peripheral subgroups is understood from context. Since a Gromov model $Y$ is a proper hyperbolic metric space, $\partial(\Gamma,\mathcal{H})$ is always compact and metrizable.

Whenever $(\Gamma,\mathcal{H})$ is nonelementary, its Bowditch boundary contains at least three points. The convergence properties of the action of $\Gamma$ on $\partial(\Gamma,\mathcal{H})$ (see below) imply that in this case, $\partial(\Gamma,\mathcal{H})$ is perfect (i.e. contains no isolated points).

If a group $\Gamma$ acts on a proper geodesic hyperbolic metric space $Y$, we can characterize the geometrical finiteness of the action entirely in terms of the topological dynamics of the action on $\partial Y$. In particular, we can understand geometrical finiteness by studying properties of convergence group actions. See [Tuk94], [Tuk98], [Bow99] for further detail on such actions, and justifications for the results stated in this section.

The name “conical limit point” makes more sense in the context of convergence group actions on boundaries of hyperbolic metric spaces.

A bounded neighborhood of a geodesic in a hyperbolic metric space looks like a “cone,” hence “conical limit.”

If $\gamma_{n}\cdot y_{0}$ limits conically to a point $z\in\partial Y$ for some (hence any) basepoint $y_{0}\in Y$, we just say that $\gamma_{n}$ limits conically to $z$.

Whenever a group $\Gamma$ acts as a convergence group on a perfect compact metrizable space $Z$, every point in $Z$ is either a conical limit point or a bounded parabolic point, and the stabilizer of each parabolic point is finitely generated, we say the $\Gamma$-action on $Z$ is geometrically finite. This is justified by a theorem of Yaman [Yam04], which says that any such group action is induced by the action of a relatively hyperbolic group on a Gromov model $Y$ whose boundary is equivariantly homeomorphic to $Z$. We can then identify the space $Z$ with the Bowditch boundary $\partial(\Gamma,\mathcal{H})$. The set of parabolic points in $Z$ coincides exactly with the set of fixed points of peripheral subgroups.

Whenever $(\Gamma,\mathcal{H})$ is a relatively hyperbolic pair, there are only finitely many conjugacy classes of groups in $\mathcal{H}$. We can fix a finite set $\mathcal{P}$ of conjugacy representatives for the groups in $\mathcal{H}$. The set $\mathcal{P}$ corresponds to a finite set $\Pi\subset\partial_{\mathrm{par}}\Gamma$ of parabolic points, such that

$$\mathcal{P}=\{\Gamma_{p}:p\in\Pi\}.$$

Then $\Pi$ contains exactly one point in each $\Gamma$-orbit in $\partial_{\mathrm{par}}\Gamma$.

The quasi-isometry class of $\mathrm{Cay}(\Gamma,S,\mathcal{P})$ is independent of the choice of generating set $S$. When $(\Gamma,\mathcal{H})$ is a relatively hyperbolic pair, $\mathrm{Cay}(\Gamma,S,\mathcal{P})$ is a hyperbolic metric space. It is not a proper metric space if $\mathcal{H}$ is nonempty. The Gromov boundary of $\mathrm{Cay}(\Gamma,S,\mathcal{P})$ is equivariantly homeomorphic to the set $\partial_{\mathrm{con}}\Gamma$ of conical limit points in $\partial(\Gamma,\mathcal{H})$.

Let $K$ be a maximal compact subgroup of the semisimple Lie group $G$, and let $X$ be the Riemannian symmetric space $G/K$. A subgroup $P\subset G$ is a parabolic subgroup if it is the stabilizer of a point in the visual boundary $\partial_{\infty}X$ of $X$. Two parabolic subgroups $P,Q$ are opposite if there is a bi-infinite geodesic $c$ in $X$ so that $P$ is the stabilizer of $c(\infty)$ and $Q$ is the stabilizer of $c(-\infty)$.

The compact homogeneous $G$-space $G/P$ is called a flag manifold. If $P$ and $Q$ are parabolic subgroups, then we say that two flags $\xi^{+}\in G/P$ and $\xi^{-}\in G/Q$ are opposite if the stabilizers of $\xi^{+}$, $\xi^{-}$ are opposite parabolic subgroups. (In particular a conjugate of $Q$ must be opposite to $P$).

Let $\mathfrak{g}$ be the Lie algebra of $G$, and let $\mathfrak{k}$ be the Lie algebra of the maximal compact $K$. We can decompose $\mathfrak{g}$ as $\mathfrak{k}\oplus\mathfrak{p}$, and fix a maximal abelian subalgebra $\mathfrak{a}\subset\mathfrak{p}$. The restriction of the Killing form $B$ to $\mathfrak{p}$ is positive definite, so any maximal abelian $\mathfrak{a}\subset\mathfrak{p}$ is naturally endowed with a Euclidean structure.

Each element of the abelian subalgebra $\mathfrak{a}$ acts semisimply on $\mathfrak{g}$, with real eigenvalues. So we let $\Sigma\subset\mathfrak{a}^{*}$ denote the set of roots for this choice of $\mathfrak{a}$, i.e. the set of nonzero linear functionals $\alpha\in\mathfrak{a}^{*}$ such that the linear map $\mathfrak{g}\to\mathfrak{g}$ given by $X-\alpha(X)I$ has nonzero kernel for every $X\in\mathfrak{a}$. We have a restricted root space decomposition

$$\mathfrak{g}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha},$$

where $X\in\mathfrak{a}$ acts on $\mathfrak{g}_{\alpha}$ by multiplication by $\alpha(X)$.

We choose a set of simple roots $\Delta\subset\Sigma$ so that each $\alpha\in\Sigma$ can be uniquely written as a linear combination of elements of $\Delta$ with coefficients either all nonnegative or all nonpositive. We let $\Sigma_{+}$ denote the positive roots, i.e. roots which are nonnegative linear combinations of elements of $\Delta$.

The simple roots $\Delta$ determine a Euclidean Weyl chamber

$$\mathfrak{a}^{+}=\{x\in\mathfrak{a}:\alpha(x)\geq 0,\textrm{ for all }\alpha\in\Delta\}.$$

The kernels of the roots $\alpha\in\Delta$ are the walls of the Euclidean Weyl chamber.

Choosing a maximal compact $K$, a maximal abelian $\mathfrak{a}\subset\mathfrak{p}$, and a Euclidean Weyl chamber $\mathfrak{a}^{+}$ determines a Cartan projection

$$\mu:G\to\mathfrak{a}^{+},$$

uniquely determined by the equation $g=k\exp(\mu(g))k^{\prime}$, where $k,k^{\prime}\in K$ and $\mu(g)\in\mathfrak{a}^{+}$.

Fix a subset $\theta$ of the simple roots $\Delta$. We define a standard parabolic subgroup $P^{+}_{\theta}$ to be the normalizer of the Lie algebra

$$\bigoplus_{\alpha\in\Sigma^{+}_{\theta}}\mathfrak{g}_{\alpha},$$

where $\Sigma^{+}_{\theta}$ is the set of positive roots which are not in the span of $\Delta-\theta$. The opposite subgroup $P^{-}$ is the normalizer of

$$\bigoplus_{\alpha\in\Sigma^{+}_{\theta}}\mathfrak{g}_{-\alpha}.$$

Every parabolic subgroup $P\subset G$ is conjugate to a unique standard parabolic subgroup $P^{+}_{\theta}$, and every pair of opposite parabolics $(P^{+},P^{-})$ is simultaneously conjugate to a unique pair $(P^{+}_{\theta},P^{-}_{\theta})$.

For a fixed $\theta\subset\Delta$, the group $P^{+}_{\theta}$ is the stabilizer of the endpoint of a geodesic ray $\exp(tZ)\cdot p$, where $p\in X$ is the image of the identity in $G/K$, and for any $\alpha\in\Delta$, the element $Z\in\mathfrak{a}^{+}$ satisfies

$$\alpha(Z)=0\iff\alpha\in\Delta-\theta.$$

An affine chart is the unique open dense orbit of $\operatorname{Stab}_{G}(\xi)$ in $G/P^{+}$. When $G=\operatorname{PGL}(d,\mathbb{R})$ and $P^{+}$ is the stabilizer of a line $\ell\subset\mathbb{R}^{d}$, $G/P^{+}$ is identified with $\mathbb{P}(\mathbb{R}^{d})$ and this notion of affine chart agrees with the usual one in $\mathbb{P}(\mathbb{R}^{d})$.

There is a close connection between $P$-divergence in the group $G$ and the topological dynamics of the action of $G$ on the associated flag manifold $G/P$. Kapovich-Leeb-Porti frame this connection in terms of a contraction property for $P$-divergent sequences.

The importance of contracting sequences is captured by the following:

3.5 implies in particular that if $g_{n}\in G$ is $P^{+}$-divergent, then up to subsequence there is an open subset $U\subset G/P^{+}$ such that $g_{n}\cdot U$ converges to a singleton in $G/P^{+}$. It turns out that this “weak contraction property” is enough to characterize $P^{+}$-divergence.

{restatable}

proplocalContractingImpliesGlobalContracting Let $g_{n}$ be a sequence in $G$, and suppose that for some nonempty open subset $U\subset G/P^{+}$, we have $g_{n}\cdot U\to\{\xi\}$ for $\xi\in G/P^{+}$. Then $g_{n}$ is $P^{+}$-divergent, and has a unique $P^{+}$-limit point $\xi\in G/P^{+}$.

We provide a proof of this fact in Appendix A.

$P$-divergent sequences are equivalent to the ${\tau_{\mathrm{mod}}}$-regular sequences discussed in the work of Kapovich-Leeb-Porti, where ${\tau_{\mathrm{mod}}}$ is the unique face corresponding to $P$ in a spherical model Weyl chamber. We explain the connection here.

For any point $p\in X$, we let $\mathfrak{p}$ be the uniquely determined subspace of $\mathfrak{g}$ such that $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, where $\mathfrak{k}$ is the Lie algebra of the stabilizer of $p$ in $G$.

Let $z\in\partial_{\infty}X$. There is a point $p\in X$, a maximal abelian subalgebra $\mathfrak{a}\subset\mathfrak{p}$, a Euclidean Weyl chamber $\mathfrak{a}^{+}\subset\mathfrak{a}$, and a unit-length $Z\in\mathfrak{a}^{+}$ such that $z$ is the endpoint of the geodesic ray $c(t)=\exp(tZ)\cdot p$.

Up to the action of the stabilizer of $z$, the point $p$, the maximal abelian subalgebra $\mathfrak{a}$, the Euclidean Weyl chamber $\mathfrak{a}^{+}$, and the unit vector $Z\in\mathfrak{a}^{+}$ are uniquely determined. In addition, the stabilizer in $G$ of the triple $(p,\mathfrak{a},\mathfrak{a}^{+})$ acts trivially on $\mathfrak{a}^{+}$.

This means that we can identify the space $\partial_{\infty}X/G$ with the set of unit vectors in any Euclidean Weyl chamber $\mathfrak{a}^{+}$. This set has the structure of a spherical simplex. We let $\sigma_{\mathrm{mod}}$ denote the model spherical Weyl chamber $\partial_{\infty}X/G$.

We let $\pi:\partial_{\infty}X\to\sigma_{\mathrm{mod}}$ be the type map to the model spherical Weyl chamber. For fixed $z\in\partial_{\infty}X$, we let $P_{z}$ denote the parabolic subgroup stabilizing $z$.

After choosing a maximal compact $K$, a maximal abelian $\mathfrak{a}\subset\mathfrak{p}$, and a Euclidean Weyl chamber $\mathfrak{a}^{+}$, the data of a face ${\tau_{\mathrm{mod}}}$ of the spherical simplex $\sigma_{\mathrm{mod}}$ is the same as the data of a subset of the simple roots of $G$: the set of roots identifies a collection of walls of the Euclidean Weyl chamber $\mathfrak{a}^{+}$. The intersection of those walls with the unit sphere in $\mathfrak{a}$ is uniquely identified with a face of $\sigma_{\mathrm{mod}}$.

For a fixed model face ${\tau_{\mathrm{mod}}}\subset\sigma_{\mathrm{mod}}$, we let $P_{\tau_{\mathrm{mod}}}$ denote any parabolic subgroup which is the stabilizer of a point $z\in\pi^{-1}({\tau_{\mathrm{mod}}})$. All such parabolic subgroups are conjugate, so as a $G$-space the flag manifold $G/P_{\tau_{\mathrm{mod}}}$ depends only on the model face ${\tau_{\mathrm{mod}}}$.

When $\rho:\Gamma\to G$ is EGF, the action of $\rho(\Gamma)$ on the boundary set $\Lambda$ is by definition an extension of the topological dynamical system $(\Gamma,\partial(\Gamma,\mathcal{H}))$. When $\Gamma$ is non-elementary, convergence dynamics imply that the homomorphism $\Gamma\to\operatorname{Homeo}(\partial(\Gamma,\mathcal{H}))$ has finite kernel and discrete image. So the map $\Gamma\to\operatorname{Homeo}(\Lambda)$ must also have discrete image and finite kernel, and therefore so does the representation $\rho:\Gamma\to G$. The case where $\Gamma$ is elementary can be verified directly.

Let $\rho:\Gamma\to G$ be an EGF representation with boundary map $\phi:\Lambda\to\partial(\Gamma,\mathcal{H})$. By assumption, we know there exists an open subset $C_{z}\subset G/P$ for each $z\in\partial(\Gamma,\mathcal{H})$, satisfying the extended convergence dynamics conditions (Definition 1.1). In general, there is not a canonical choice for the set $C_{z}$. We are able to make some assumptions about the properties of the $C_{z}$, however.

It is often possible to prove properties of relatively hyperbolic groups by first showing that the property holds for conical subsequences in the group, and then showing that the property holds inside of peripheral subgroups. There is a characterization of the EGF property along these lines, which is frequently useful for constructing examples of EGF representations (see [Wei23b]).

{restatable}

propconicalPeripheralEGF Let $\rho:\Gamma\to G$ be a representation of a relatively hyperbolic group, and let $\Lambda\subset G/P$ be a closed $\rho(\Gamma)$-invariant set, where $P\subset G$ is a symmetric parabolic subgroup. Suppose that $\phi:\Lambda\to\partial(\Gamma,\mathcal{H})$ is a continuous surjective $\rho$-equivariant antipodal map.

Then $\rho$ is an EGF representation with EGF boundary extension $\phi$ if and only if both of the following conditions hold:

1. (a)

   For any sequence $\gamma_{n}\in\Gamma$ limiting conically to some point in $\partial(\Gamma,\mathcal{H})$, $\rho(\gamma_{n}^{\pm 1})$ is $P$-divergent and every $P$-limit point of $\rho(\gamma_{n}^{\pm 1})$ lies in $\Lambda$.

2. (b)

   For every parabolic point $p\in\partial_{\mathrm{par}}(\Gamma,\mathcal{H})$, there exists an open set $C_{p}\subset G/P$, with $\Lambda-\phi^{-1}(p)\subset C_{p}$, such that for any compact $K\subset C_{p}$ and any open set $U$ containing $\phi^{-1}(p)$, for all but finitely many $\gamma\in\Gamma_{p}$, we have $\rho(\gamma)\cdot K\subset U$.

The proof of Section 4.3 requires the technical machinery of relative quasigeodesic automata, so we defer it to Section 8. At the end of Section 8, we also provide another (weaker) characterization of EGF representations which may be of interest.