Finiteness of CAT $(0)$ group actions

Nicola Cavallucci and Andrea Sambusetti

Abstract.

We prove some finiteness results for discrete isometry groups $\Gamma$ of uniformly packed CAT $(0)$ -spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT $(0)$ -orbispaces $M=\Gamma\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $\Gamma$ is bounded above by a universal constant only depending on the packing constants and the codiameter. The main tool is a splitting theorem for sufficiently collapsed actions: namely we show that if a geodesically complete, packed, CAT $(0)$ -space admits a discrete, cocompact group of isometries with sufficiently small systole then it necessarily splits a non-trivial Euclidean factor.

N. Cavallucci has been partially supported by the SFB/TRR 191, funded by the DFG

A. Sambusetti is member of GNSAGA and acknowledges the support of INdAM during the preparation of this work.

1. Introduction

This is the first of two papers devoted to the theory of convergence for groups $\Gamma$ acting geometrically (that is discretely, by isometries and with compact quotient) on CAT $(0)$ -spaces $X$ . In this one, we will mainly focus on finiteness and splitting results. In our work, all CAT $(0)$ -spaces $X$ are assumed to be proper and geodesically complete, which ensures many desirable geometric properties, such as the equality of topological dimension and Hausdorff dimension, the existence of a canonical measure $\mu_{X}$ , etc. (see [Kle99], [LN19] and Section 2 for fundamentals on CAT $(0)$ -spaces).

The following is the first main finding of this paper:

Theorem A (Finiteness).

Given $P_{0},r_{0},D_{0}>0$ , there exist only finitely many groups $\Gamma$ acting discretely and nonsingularly by isometries on some proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space with quotient of diameter at most $D_{0}$ , up to isomorphism of abstract groups.
(We stress the fact that, in the above theorem, the CAT $(0)$ -spaces on which the groups $\Gamma$ are supposed to act are not fixed a-priori).

Here is a quick explanation of the terms used in the statement above.
We say that a group $\Gamma$ acts nonsingularly on $X$ if there exists at least one point $x\in X$ such that $\text{Stab}_{\Gamma}(x)$ is trivial (in particular, the action of $\Gamma$ on $X$ is faithful). This condition has many natural consequences (e.g., the existence of a fundamental domain) and is a mild assumption on the action, which is automatically satisfied for instance when the action is faithful and either the group is torsion-free or $X$ is a homology manifold (see Section 3 and [CS22]). We stress the fact that nonsingularity is essential for Theorem A: in [BK90], Bass and Kulkarni exhibit an infinite family of non-isomorphic, discrete groups $\Gamma_{j}$ acting (singularly) on the same CAT $(0)$ space $X$ (a regular tree with bounded valency) with diam $(\Gamma_{j}\backslash X)\leq D_{0}$ for all $j$ ; we thank P.E. Caprace for pointing out to us this example (see Remark 5.3). We will say that $\Gamma$ acts $D_{0}$ -cocompactly on $X$ if diam $(\Gamma\backslash X)\leq D_{0}$ .

On the other hand, a metric space $X$ is $(P_{0},r_{0})$ -packed if all balls $B_{X}(x,3r_{0})$ of radius $3r_{0}$ in $X$ contain at most $P_{0}$ points that are $2r_{0}$ -separated. This condition should be thought as weak form of lower curvature bound at scale $r_{0}$ ; however, it is much weaker than assuming the curvature bounded below in the sense of Alexandrov, or a lower bound on the Ricci curvature for Riemanniann manifolds, or the CD $(k,n)$ condition for metric measure spaces. Indeed, the Bishop-Gromov’s comparison theorem for Riemannian $n$ -manifolds with Ric ${}_{X}\geq-(n-1)k$ (or its generalization to CD $(k,n)$ spaces) yields a doubling condition

(1)

\frac{\mu(B_{X}(x,2r))}{\mu(B_{X}(x,r))}\leq C(n,k,R)\hskip 14.22636pt

for the measure of all $r$ -balls with $r\leq R$ (for the Riemannian measure in case of manifolds, and for the reference measure in case of CD $(k,n)$ -spaces, cp. for instance [Stu06]), from which the packing condition at scale $r_{0}\leq r/4$ easily follows. See [BCGS17] for a detailed comparison of packing, doubling, entropy and curvature conditions. Let us just recall here that the packing condition for a CAT $(0)$ -space $X$ is also strictly weaker than the doubling condition (1) for the natural measure $\mu_{X}$ since a local doubling condition implies that the space has pure dimension (i.e. the tangent cones at all points $x$ have the same geometric dimension $n$ , see [CS21, Theorem C]), which restricts considerably the class of spaces under consideration.

Moreover, for the spaces we are interested in, a packing condition is a natural, and in some sense minimal, assumption. Indeed, every metric space $X$ with a compact quotient is $(P_{0},r_{0})$ -packed for suitable constants $P_{0},r_{0}$ (cp. [Cav22a, Proof of Lemma 5.4]). Namely, let $\textup{CAT}_{0}(D_{0})$ be the class of all isometric actions $\Gamma\curvearrowright X$ where $X$ is a proper, geodesically complete, CAT $(0)$ -space, and $\Gamma$ is a $D_{0}$ -cocompact, discrete subgroup of $\text{Isom}(X)$ ; then the $(P_{0},r_{0})$ -packing conditions define a filtration

\textup{CAT}_{0}(D_{0})=\bigcup_{P_{0},r_{0}}\textup{CAT}_{0}(P_{0},r_{0},D_{0})

where $\textup{CAT}_{0}(P_{0},r_{0},D_{0})$ is the subset of $\textup{CAT}_{0}(D_{0})$ made of the actions $\Gamma\curvearrowright X$ such that $X$ is, moreover, $(P_{0},r_{0})$ -packed.

As proved in [CS21], for geodesically complete CAT $(0)$ -spaces, a packing condition at some scale $r_{0}$ is equivalent to a uniform upper bound of the canonical measure of all $r$ -balls (a condition sometimes called macroscopic scalar curvature bounded below cp. [Gut10], [Sab20]). Namely, there exist functions $v,V\colon(0,+\infty)\to(0,+\infty)$ depending only on $P_{0},r_{0}$ such that for all $x\in X$ and $R>0$ we have (cp. Proposition 2.3):

(2)

v(R)\leq\mu_{X}(B(x,R))\leq V(R).

Moreoever, for geodesically complete CAT $(0)$ -spaces the $(P_{0},r_{0})$ -packing condition yields an important generalization of the classical Margulis’ Lemma, due to Breuillard-Green-Tao: there exists a constant $\varepsilon_{0}(P_{0},r_{0})\!>0$ such that for every discrete group of isometries $\Gamma$ of $X$ the $\varepsilon_{0}$ -almost stabilizer $\Gamma_{\varepsilon_{0}}(x)$ of any point $x$ is virtually nilpotent (cp. [BGT11] and Section 2.2 for details). We call this $\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})$ the Margulis constant, since it plays the role of the classical Margulis constant in our metric setting.

We will see that the finiteness up to isomorphism of Theorem A can be improved to finiteness up to equivariant homotopy equivalence of the pairs $(X,\Gamma)$ (see Section 5.2 for the precise definition), and then deduce from Theorem A corresponding finiteness results for the quotient spaces $M=\Gamma\backslash X$ . We will call such spaces CAT $(0)$ -orbispaces ¹¹1When restricting our attention to groups $\Gamma$ acting rigidly on $X$ (i.e. such that every $g\in\Gamma$ acting as the identity on an open subset is trivial) then this definition is equivalent to the notion of rigid, developable orbispace as defined in [DLHG90, Ch.11] (with CAT $(0)$ universal covering in the sense of orbispaces). However, our results apply to all nonsingular CAT $(0)$ -orbispaces., using this term as a shortening for quotient of a (proper, geodesically complete) CAT $(0)$ -space $X$ by a discrete, isometry group $\Gamma$ ; we will also say that the CAT $(0)$ -orbispace $M$ is nonsingular if $\Gamma$ acts nonsingularly on $X$ . Notice that if $\Gamma$ is torsion-free then $M$ is a locally CAT $(0)$ -space.
For the following, let us define the class of all CAT $(0)$ -orbispaces $M=\Gamma\backslash X$

{\mathcal{O}}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})

where $\Gamma\curvearrowright X$ is in CAT ${}_{0}(P_{0},r_{0},D_{0})$ , and the subclasses

{\mathcal{L}}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})\hskip 2.84526pt\subset\hskip 2.84526pt{\mathcal{NS}}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})

of those which are, respectively, locally CAT $(0)$ -spaces and non-singular.
Then we obtain:

Corollary B.

There are finitely many spaces in the class ${\mathcal{L}}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})$ up to homotopy equivalence, and finitely many spaces in ${\mathcal{NS}}\textup{-CAT}_{0}(P_{0},r_{0},D_{0})$ , up to equivariant homotopy equivalence.
(Two orbispaces $M=\Gamma\backslash X$ and $M^{\prime}=\Gamma^{\prime}\backslash X^{\prime}$ are equivariantly homotopy equivalent if there exists a group isomorphism $\varphi:\Gamma\rightarrow\Gamma^{\prime}$ and a $\varphi$ -equivariant homotopy equivalence $F:X\rightarrow X^{\prime}$ , that is such that $F(g\cdot x)=\varphi(g)\cdot F(x)$ for all $g\in\Gamma$ and all $x\in X$ .)

A particular case of Theorem A, declined in the Riemannian setting, is:

Corollary C.

For every fixed $n,\kappa,D_{0}$ , there exist only finitely many compact $n$ -dimensional Riemannian orbifolds $M$ with curvature $-\kappa^{2}\leq k(M)\leq 0$ and $\textup{diam}(M)\leq D_{0}$ , up to equivariant diffeomorphisms.
(Two Riemannian orbifolds $M=\Gamma\backslash X$ , $M^{\prime}=\Gamma^{\prime}\backslash X^{\prime}$ are equivariantly diffeomorphic if there exists a diffeomorphism $X\rightarrow X^{\prime}$ which is equivariant with respect to some isomorphism $\varphi:\Gamma\rightarrow\Gamma^{\prime}$ .)

In particular, for torsion-free orbifolds, this gives a new proof of the finiteness of compact Riemannian $n$ -manifolds $M$ with curvature $-\kappa^{2}\leq k(M)\leq 0$ and diam $(M)\leq D_{0}$ , modulo diffeomorphisms (this result was announced without proof by Gromov in [Gro78] and proved by Buyalo, up to homeomorphisms, in [Buy83]; the orbifold version was proved by Fukaya [Fuk86] in strictly negative curvature only).

Finiteness theorems in the spirit of Corollary C have been proved in different contexts in literature, the archetype of all of them being of course Weinstein’s theorem for pinched, positively curved manifolds [Wei67] and Cheeger’s finiteness theorems in bounded sectional curvature of variable sign (see [Che70], [Gro07], [Pet84], [Yam85]). In all these theorems, the finiteness is obtained by assuming a positive lower bound on the injectivity radius, or deducing such a lower bound from the combination of geometric and topological assumptions (see for instance [PT99] for simply connected manifolds with finite second homotopy groups).

A result similar to Theorem A was recently proved in [BCGS21], where the authors obtain the finiteness of torsionless groups $\Gamma$ acting faithfully, discretely and $D_{0}$ -cocompactly on $\delta_{0}$ -hyperbolic spaces with entropy $\leq H_{0}$ . They achieve this by proving a positive, universal lower bound of the systole of $\Gamma$ , similar to the classical Heintze-Margulis’ Lemma (holding for manifolds with pinched, strictly negative curvature). Namely, for a discrete isometry group $\Gamma$ of $X$ let us call, respectively,

\text{sys}(\Gamma,x):=\inf_{g\in\Gamma\setminus\{\text{id}\}}d(x,gx)

\text{sys}(\Gamma,X):=\inf_{x\in X}\text{sys}(\Gamma,x)

the systole of $\Gamma$ at $x$ and the (global) systole of $\Gamma$ (notice that the systole of the fundamental group $\Gamma=\pi_{1}(M)$ of a nonpositively curved manifold $M$ , acting on its Riemannian universal covering $X$ , precisely equals twice the injectivity radius of $M$ ). Then, in [BCGS21], the authors prove that $\text{sys}(\Gamma,X)$ is bounded below by a positive constant $c_{0}(\delta_{0},H_{0},D_{0})$ only depending on the hyperbolicity constant, the entropy of $X$ and on the diameter of $\Gamma\backslash X$ .This bound is, clearly, consequence of the Gromov hyperbolicity of the space $X$ , which is a form of strictly negative curvature at macroscopic scale. In contrast, for the group actions in the classes considered in Theorem A and in the above corollaries, the systole may well be arbitrarily small, since the curvature is only assumed to be non-positive. The main difficulty in Theorem A and in the corollaries above precisely boils down in understanding what happens when the systole or the injectivity radius tend to zero.

In fact, the problem of collapsing will be of primary interest in this work. Recall that a Riemannian manifold $M$ is called $\varepsilon$ -collapsed if the injectivity radius is smaller than $\varepsilon$ at every point. The theory of collapsing for Riemannian manifolds with bounded sectional curvature was developed by Cheeger, Gromov and Fukaya: for a differentiable manifold $M$ , the existence of a Riemannian metric with bounded sectional curvature sufficiently collapsed imposes strong restrictions to its topology. Namely, there exists an $\varepsilon_{0}(n)>0$ such that if a Riemannian $n$ -manifold with $|K_{M}|\leq 1$ is $\varepsilon$ -collapsed with $\varepsilon<\varepsilon_{0}(n)$ , then $M$ admits a so-called $F$ -structure of positive rank, cp. [CG86]-[CG90] and [CFG92] (see also the works of Fukaya [Fuk87]-[Fuk88] for collapsible manifolds with uniformly bounded diameter).

More specifically about nonpositively curved geometry, Buyalo [Buy90a]-[Buy90b] first, in dimension smaller than $5$ , and Cao-Cheeger-Rong [CCR01] later, in any dimension, studied the possibility of collapsing compact, $n$ -dimensional manifolds $M$ with bounded sectional curvature $-\kappa^{2}\leq K_{M}\leq 0$ . They proved that either the injectivity radius at some point is bounded below by a universal positive constant $i_{0}(n,\kappa)>0$ , or $M$ admits a so-called abelian local splitting structure: this is, roughly speaking, a decomposition of the universal covering $X$ into a union of minimal sets of hyperbolic isometries with the additional property that if two minimal sets intersect then the corresponding isometries commute. A prototypical example of collapsing with bounded, non-positive curvature is the following, which might be useful to have in mind for the sequel (cp. [Gro78], Section 5 and [Buy81], Section 4): consider two copies $\Sigma_{1},\Sigma_{2}$ of the same hyperbolic surface with connected, geodesic boundary of length $\frac{1}{n}$ , then take the products $\Sigma^{\prime}_{i}=\Sigma_{i}\times S^{1}$ with a circle of length $\frac{1}{n}$ , and glue $\Sigma^{\prime}_{1}$ to $\Sigma^{\prime}_{2}$ by means of an isometry $\varphi$ of the boundaries $\partial\Sigma^{\prime}_{i}=\partial\Sigma_{i}\times S^{1}$ which interchanges the circles $\partial\Sigma_{i}$ with $S^{1}$ . This yields a nonpositively curved $3$ -manifold $M_{n}$ (a graph manifold) with sectional curvature $-1\leq K_{M_{n}}\leq 0$ , whose injectivity radius at every point is arbitrarily small provided that $n\gg 0$ .

Coming back to our metric setting, where we also allow groups with torsion and isometries with fixed points, it is useful to distinguish between systole and free systole: we define the free systole of $\Gamma$ at $x$ and the (global) free systole of $\Gamma$ respectively as

\text{sys}^{\diamond}(\Gamma,x)=\inf_{g\in\Gamma^{\diamond}}d(x,gx)

\text{sys}^{\diamond}(\Gamma,X)=\inf_{x\in X}\text{sys}^{\diamond}(\Gamma,x)

where $\Gamma^{\diamond}$ denotes the subset of torsion-free elements of $\Gamma$ .
The first non-trivial finding for non-singular actions is that, when assuming a bound on the diameter, then the smallness of the systole at every point is quantitatively equivalent to the smallness of the global free systole (see Theorem 3.1). Therefore, we will say that a $D_{0}$ -cocompact action of a group $\Gamma$ on a CAT $(0)$ -space $X$ (or, equivalently, the quotient space $M=\Gamma\backslash X$ ) is $\varepsilon$ -collapsed if $\textup{sys}^{\diamond}(\Gamma,X)\leq\varepsilon$ .

The following theorem, which is the key to our finiteness theorems, shows that if $X$ admits a discrete, $D_{0}$ -cocompact action which is $\varepsilon$ -collapsed for sufficiently small $\varepsilon$ , then $X$ necessarily splits a non-trivial Euclidean factor:

Theorem D (Splitting of an Euclidean factor under $\varepsilon$ -collapsed actions).

Let $P_{0},r_{0},D_{0}>0$ . There exists $\sigma_{0}=\sigma_{0}(P_{0},r_{0},D_{0})>0$ such that if $X$ is a proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space admitting a discrete, $D_{0}$ -cocompact group of isometries $\Gamma$ with $\textup{sys}^{\diamond}(\Gamma,X)\leq\sigma_{0}$ then $X$ splits isometrically as $Y\times\mathbb{R}^{k}$ , with $k\geq 1$ .

Notice that, as the prototypical example above shows already for manifolds, the splitting of $X$ does not hold without an a priori bound on the diameter.

As proved by Buyalo in [Buy83], when $M=\!\Gamma\!\backslash X$ is a manifold with pinched, nonpositive sectional curvature $-1\leq k_{M}\leq\!0$ and diameter bounded by $D$ , then there exists a positive constant $\varepsilon(n,D)$ such that the condition of being $\varepsilon$ -collapsed for $\varepsilon<\varepsilon(n,D)$ implies the existence of a normal, free abelian subgroup $A$ of rank $k\geq 1$ ; then, $\Gamma$ virtually splits $A$ , and the existence of a non-trivial Euclidean factor for $X$ can be deduced from classical splitting theorems for non-positively curved manifolds (see [Ebe88], or [BH13, Prop.6.23]). The proof of Buyalo uses a stability property of hyperbolic isometries based on the fact that two isometries which coincide on an open subset are equal, a fact which is drastically false in our metric context (see the discussion in [CS22, Example 1.2]). Also, notice that in Theorem D we do not assume any bound on the order of torsion elements $g\in\Gamma$ (as opposite to [Fuk86]): actually, bounding the order of the torsion elements of $\Gamma$ is one of the main conclusions of this work, see Corollary F below.
Our proof is more inspired to [CM09b]-[CM19]: we do not prove the existence of a normal free abelian subgroup, rather we find a free abelian, commensurated subgroup $A^{\prime}$ of $\Gamma$ , from which we construct a $\Gamma$ -invariant closed convex subset $X_{0}\subset X$ which splits as $Y\times{\mathbb{R}}^{k}$ , and then we use the minimality of the action to deduce that $X_{0}=X$ (see Section 4 for the proof). Actually, in our metric setting it can happen that $\Gamma$ does not have any non-trivial, normal, free abelian subgroup at all (see [CS23]).

The final step for Theorem A is realizing that the $\varepsilon$ -collapsing of an action of $\Gamma$ on $X$ can occur on different subspaces of $X$ at different scales; a careful analysis of this phenomenon allows us to renormalize, in a precise sense, the metric of $X$ , keeping the diameter of the quotient $\Gamma\backslash X$ bounded, and obtaining the following result, similar to Buyalo’s [Buy83, Theorem 1.1], which we believe is of independent interest (see Section 5 and Remark 5.2 for a more precise statement):

Theorem E (Renormalization).

Given $P_{0},r_{0},D_{0}$ , there exist $s_{0}=s_{0}(P_{0},r_{0},D_{0})>0$ and $\Delta_{0}=\Delta_{0}(P_{0},D_{0})$ such that the following holds. Let $\Gamma$ be a discrete and $D_{0}$ -cocompact isometry group of a proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space $X$ : then, $\Gamma$ admits also a faithful, discrete, $\Delta_{0}$ -cocompact action by isometries on a $\textup{CAT}(0)$ -space $X^{\prime}$ isometric to $X$ , such that $\textup{sys}^{\diamond}(\Gamma,X^{\prime})\geq s_{0}$ .
Moreover, the action of $\Gamma$ on $X$ is nonsingular if and only if the action on $X^{\prime}$ is nonsingular.

This theorem allows us to deduce the finiteness of nonsinngular groups $\Gamma$ in the class CAT ${}_{0}(P_{0},r_{0},D_{0})$ , using Serre’s classical presentation of groups acting on simply connected spaces, and to obtain Corollary C from Cheeger’s and Fukaya’s finiteness theorems.

Remark. Explicitly, we may take $\Delta_{0}=2^{n_{0}}(D_{0}+\sqrt{n_{0}})$ , where $n_{0}=P_{0}/2$ bounds the dimension of every $X$ in CAT ${}_{0}(P_{0},r_{0},D_{0})$ (see Proposition 2.3). Also the constant $s_{0}$ could be made explicit in terms of $P_{0},R_{0},D_{0}$ and of the Margulis’ constant $\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})$ , following the proof of Theorem E and using the function $\sigma_{0}(P_{0},r_{0},D_{0})$ appearing in Theorem D (itself explicitable in terms of $P_{0},R_{0},D_{0}$ and of the universal bound of the index of the lattice of translations ${\mathcal{L}}(G)$ in any crystallographic group $G$ of dimension $\leq n_{0}$ ). The only quantity which is not explicit is the Margulis constant $\varepsilon_{0}(P_{0},r_{0})$ provided by [BGT11].

Another immediate (although a-priori highly non-trivial) consequence of Theorem A is that there exists a uniform bound on the order of the finite subgroups of the groups $\Gamma$ under consideration. We stress that the existence of such a bound is new also for isometry groups $\Gamma$ of Hadamard manifolds; for instance, this is a key-assumption in the theory of convergence of Riemannian orbifolds of Fukaya. A similar bound is proved in [Fuk86] under the additional assumption of a lower bound on the volume of $M=\Gamma\backslash X$ . More explicitly, there exists a constant $b_{0}=b_{0}(P_{0},r_{0})>0$ (whose geometric meaning is explained in Proposition 3.2) such that the following holds:

Corollary F (Universal bound of the order of finite subgroups).

Let $P_{0},r_{0},D_{0}\!>0$ . Then for every nonsingular $\Gamma\curvearrowright X$ in CAT ${}_{0}(P_{0},r_{0},D_{0})$ , every finite subgroup $F<\Gamma$ has order

|F|\leq\frac{V(\Delta_{0})}{v\left(\frac{1}{2}b_{0}s_{0}\right)}

(where $v,V$ are the universal functions appearing in (2), and $\Delta_{0},s_{0}$ are the constants of Theorem E).

Acknowledgments. The authors thank P.E.Caprace, S.Gallot and A.Lytchak for many interesting discussions during the preparation of this paper, and D.Semola for pointing us to interesting references.

Notation. Throughout the paper, we will adopt the following convention: a letter with the subscript ₀ will always denote a fixed constant or a universal function depending only on the parameters $P_{0},r_{0}$ and, possibly, on $D_{0}$ . For instance, in the above theorems, we used the constants $\sigma_{0},s_{0},\Delta_{0},b_{0}$ with this meaning.

2. Preliminaries on CAT $(0)$ -spaces

We fix here some notation and recall some facts about CAT $(0)$ -spaces.
Throughout the paper $X$ will be a proper metric space with distance $d$ . The open (resp. closed) ball in $X$ of radius $r$ , centered at $x$ , will be denoted by $B_{X}(x,r)$ (resp. $\overline{B}_{X}(x,r)$ ); we will often drop the subscript $X$ when the space is clear from the context.
A geodesic in a metric space $X$ is an isometry $c\colon[a,b]\to X$ , where $[a,b]$ is an interval of $\mathbb{R}$ . The endpoints of the geodesic $c$ are the points $c(a)$ and $c(b)$ ; a geodesic with endpoints $x,y\in X$ is also denoted by $[x,y]$ . A geodesic ray is an isometry $c\colon[0,+\infty)\to X$ and a geodesic line is an isometry $c\colon\mathbb{R}\to X$ . A metric space $X$ is called geodesic if for every two points $x,y\in X$ there is a geodesic with endpoints $x$ and $y$ .

A metric space $X$ is called CAT $(0)$ if it is geodesic and every geodesic triangle $\Delta(x,y,z)$ is thinner than its Euclidean comparison triangle $\overline{\Delta}(\bar{x},\bar{y},\bar{z})$ : that is, for any couple of points $p\in[x,y]$ and $q\in[x,z]$ we have $d(p,q)\leq d(\bar{p},\bar{q})$ where $\bar{p},\bar{q}$ are the corresponding points in $\overline{\Delta}(\bar{x},\bar{y},\bar{z})$ (see for instance [BH13] for the basics of CAT(0)-geometry). As a consequence, every CAT $(0)$ -space is uniquely geodesic: for every two points $x,y$ there exists a unique geodesic with endpoints $x$ and $y$ .

A CAT $(0)$ -metric space $X$ is geodesically complete if evvery geodesic $c\colon[a,b]\to X$ can be extended to a geodesic line. For instance, if a CAT $(0)$ -space is a homology manifold then it is always geodesically complete, see [BH13, Proposition II.5.12]).

The boundary at infinity of a CAT $(0)$ -space $X$ (that is, the set of equivalence classes of geodesic rays, modulo the relation of being asymptotic), endowed with the Tits distance, will be denoted by $\partial X$ , see [BH13, Chapter II.9].

A subset $C$ of $X$ is said to be convex if for all $x,y\in C$ the geodesic $[x,y]$ is contained in $C$ . Given a subset $Y\subseteq X$ we denote by $\text{Conv}(Y)$ the convex closure of $Y$ , that is the smallest closed convex subset containing $Y$ .If $C$ is a convex subset of a CAT $(0)$ -space $X$ then it is itself CAT $(0)$ , and its boundary at infinity $\partial C$ naturally and isometrically embeds in $\partial X$ .

We will denote by HD $(X)$ and TD $(X)$ the Hausdorff and the topological dimension of a metric space $X$ , respectively. By [LN19] we know that if $X$ is a proper and geodesically complete CAT $(0)$ -space then every point $x\in X$ has a well defined integer dimension in the following sense: there exists $n_{x}\in\mathbb{N}$ such that every small enough ball around $x$ has Hausdorff dimension equal to $n_{x}$ . This defines a stratification of $X$ into pieces of different integer dimensions: namely, if $X^{k}$ denotes the subset of points of $X$ with dimension $k$ , then

X=\bigcup_{k\in\mathbb{N}}X^{k}.

The dimension of $X$ is the supremum of the dimensions of its points: it coincides with the topological dimension of $X$ , cp. [LN19, Theorem 1.1].

Calling $\mathcal{H}^{k}$ the $k$ -dimensional Hausdorff measure, the formula

\mu_{X}:=\sum_{k\in\mathbb{N}}\mathcal{H}^{k}\llcorner X^{k}

defines a canonical measure on $X$ which is locally positive and locally finite.

2.1. Discrete isometry groups

Let $\text{Isom}(X)$ be the group of isometries of $X$ , endowed with the compact-open topology: as $X$ is proper, it is a topological, locally compact group.
The translation length of $g\in\text{Isom}(X)$ is by definition $\ell(g):=\inf_{x\in X}d(x,gx).$ When the infimum is realized, the isometry $g$ is called elliptic if $\ell(g)=0$ and hyperbolic otherwise. The minimal set of $g$ , $\text{Min}(g)$ , is defined as the subset of points of $X$ where $g$ realizes its translation length; notice that if $g$ is elliptic then $\text{Min}(g)$ is the subset of points fixed by $g$ . An isometry is called semisimple if it is either elliptic or hyperbolic; a subgroup $\Gamma$ of Isom $(X)$ is called semisimple if all of its elements are semisimple.
Let $\Gamma$ be a subgroup of Isom $(X)$ . For $x\in X$ and $r\geq 0$ we set

(3)

\overline{\Sigma}_{r}(x,X):=\{g\in\Gamma\text{ s.t. }d(x,gx)\leq r\}

(4)

\overline{\Gamma}_{r}(x,X):=\langle\overline{\Sigma}_{r}(x,X)\rangle

When the context is clear we will simply write $\overline{\Sigma}_{r}(x)$ and $\overline{\Gamma}_{r}(x)$ .
The subgroup $\Gamma$ is discrete if it is discrete as a subset of Isom(X) (with respect to the compact-open topology). Since $X$ is assumed to be proper and $\Gamma$ acts by isometries, this condition is the same as asking that the orbit $\Gamma x$ is discrete and $\text{Stab}_{\Gamma}(x)$ is finite for some (or, equivalently, for all) $x\in X$ . This is in turn equivalent to asking that the sets $\overline{\Sigma}_{r}(x)$ are finite for all $x\in X$ and all $r\geq 0$ .
When dealing with isometry groups $\Gamma$ of CAT $(0)$ -spaces with torsion, a difficulty is that there may exist nontrivial elliptic isometries which act as the identity on open sets. Following [DLHG90, Chapter 11], a subgroup $\Gamma$ of Isom $(X)$ will be called rigid (or slim, with the terminology used in [CS22]) if for all $g\in\Gamma$ the subset $\text{Fix}(g)$ has empty interior. Every torsion-free group is trivially rigid, as well as any discrete group acting on a CAT $(0)$ -homology manifold, as proved in [CS22, Lemma 2.1].
A CAT $(0)$ -orbispace (in the sense of [Fuk86]) is the quotient $M=\Gamma\backslash X$ of a (proper, geodesically complete) CAT $(0)$ -space $X$ by a discrete isometry group $\Gamma$ . One might define the notion of orbispace $M$ in terms of an orbifold atlas, that is a collection of uniformizing charts $\pi_{i}:V_{i}\rightarrow U_{i}\subset M$ covering $M$ (where $V_{i}$ is a locally compact space endowed with the action of a finite group $\Gamma_{i}$ such that $\pi_{i}$ induces a homeomorphism $\Gamma_{i}\backslash V_{i}\simeq U_{i}$ ) and a pseudogroup of local homeomorphisms given by changing charts. This is for instance the approach of Haefliger in [DLHG90, Ch.11], where rigid orbispaces are defined; rigid here means that the actions of the groups $\Gamma_{i}$ on $V_{i}$ are all supposed to be rigid. Every quotient of a CAT $(0)$ -space $X$ by a discrete and rigid group $\Gamma$ has a structure of rigid orbispace; reciprocally, every rigid orbispace $M$ with nonpositive curvature (that is, such that the domains $V_{i}$ of the uniformizing charts are locally CAT $(0)$ -spaces) is developable, which means that $M=\Gamma\backslash X$ , for some rigid, discrete isometry group $\Gamma$ acting on a suitable CAT $(0)$ -space $X$ , see [DLHG90, Ch.11, Théorème 8].
A group $\Gamma<\textup{Isom}(X)$ is said to be cocompact if the quotient metric space $\Gamma\backslash X$ is compact; in this case, we call codiameter of $\Gamma$ the diameter of the quotient, and we will say that $\Gamma$ is $D_{0}$ -cocompact if it has codiameter at most $D_{0}$ . Notice that the codiameter of $\Gamma$ coincides with

(5)

\inf\{r>0\text{ s.t. }\Gamma\cdot\overline{B}(x,r)=X\,\,\,\,\forall x\in X\}.

It is well-known, and we will consistently use it in the paper, that if $X$ is geodesic and $\Gamma<\textup{Isom}(X)$ is discrete and $D_{0}$ -cocompact, then for all $D\geq D_{0}$ the subset $\overline{\Sigma}_{2D}(x)$ is a generating set for $\Gamma$ , that is $\overline{\Gamma}_{2D}(x)=\Gamma$ , for every $x\in X$ ; we call this a $2D$ -short generating set of $\Gamma$ at $x$ .
Moreover, if $X$ is simply connected, then $\Gamma$ admits a finite presentation as

\Gamma=\langle\overline{\Sigma}_{2D}(x)\hskip 2.84526pt|\hskip 2.84526pt{\mathcal{R}}_{2D}(x)\rangle

where ${\mathcal{R}}_{2D}(x)$ is a subset of words of length $3$ on $\overline{\Sigma}_{2D}(x)$ , see [Ser03, App., Ch.3]. For later use we also record the following general fact.

Lemma 2.1.

Let $X$ be a geodesic metric space and let $\Gamma<\textup{Isom}(X)$ be a discrete, $D_{0}$ -cocompact group. If $\Gamma^{\prime}$ is normal subgroup of $\Gamma$ with index $[\Gamma:\Gamma^{\prime}]\leq J$ , then $\Gamma^{\prime}$ is at most $2D_{0}(J+1)$ -cocompact.

Proof.

As $\Gamma^{\prime}$ is discrete, the space $Y=\Gamma^{\prime}\backslash X$ is geodesic and the group $\Gamma^{\prime}\backslash\Gamma$ acts by isometries on it with codiameter at most $D_{0}$ . Take arbitrary $y,y^{\prime}\in Y$ and connect them by a geodesic $c$ . Let $t_{k}=k\cdot(2D_{0}+\varepsilon)$ , as far as $c(t_{k})$ is defined for integers $k$ , and let $g_{k}\in\Gamma^{\prime}\backslash\Gamma$ be such that $d(c(t_{k}),g_{k}y)\leq D_{0}$ . By construction, the points $\{g_{k}y\}$ are all distinct since they are all $\varepsilon$ -separated, so the $g_{k}$ ’s are distinct. Since $\Gamma^{\prime}\backslash\Gamma$ has cardinality at most $J$ , we conclude that $d(y,y^{\prime})\leq(J+1)\cdot(2D_{0}+\varepsilon)$ . By the arbitrariness of $\varepsilon$ , $y$ and $y^{\prime}$ we deduce that the diameter of $Y$ is at most $2D_{0}(J+1)$ . ∎

In general the full isometry group Isom $(X)$ of a CAT $(0)$ -space is not a Lie group, for instance in the case of regular trees. When a CAT $(0)$ -space $X$ admits a cocompact, discrete group of isometries $\Gamma$ then Isom $(X)$ is known to have more structure, as proved by P.-E.Caprace and N.Monod.

Proposition 2.2 ([CM09b, Thm.1.6 & Add.1.8], [CM09a, Cor.3.12]).

Let $X$ be a proper, geodesically complete, $\textup{CAT}(0)$ -space, admitting a discrete, cocompact group of isometries. Then $X$ splits isometrically as $M\times\mathbb{R}^{n}\times N$ , where $M$ is a symmetric space of noncompact type and ${\mathcal{D}}:=\textup{Isom}(N)$ is totally disconnected. Moreover

\textup{Isom}(X)\cong{\mathcal{S}}\times{\mathcal{E}}_{n}\times{\mathcal{D}}

where ${\mathcal{S}}$ is a semi-simple Lie group with trivial center and without compact factors and ${\mathcal{E}}_{n}\cong\textup{Isom}(\mathbb{R}^{n})$ .

2.2. The packing condition and Margulis’ Lemma

Let $X$ be a metric space and $r>0$ . A subset $Y$ of $X$ is called $r$ -separated if $d(y,y^{\prime})>r$ for all $y,y^{\prime}\in Y$ . Given $x\in X$ and $0<r\leq R$ we denote by Pack $(\overline{B}(x,R),r)$ the maximal cardinality of a $2r$ -separated subset of $\overline{B}(x,R)$ . Moreover we denote by Pack $(R,r)$ the supremum of Pack $(\overline{B}(x,R),r)$ among all points of $X$ . Given $P_{0},r_{0}>0$ we say that $X$ is $P_{0}$ -packed at scale $r_{0}$ (or $(P_{0},r_{0})$ -packed, for short) if Pack $(3r_{0},r_{0})\leq P_{0}$ . We will simply say that $X$ is packed if it is $P_{0}$ -packed at scale $r_{0}$ for some $P_{0},r_{0}>0$ .
The packing condition should be thought as a metric, weak replacement of a Ricci curvature lower bound: for more details and examples see [CS21]. Actually, by Bishop-Gromov’s Theorem, for a $n$ -dimensional Riemannian manifold a lower bound on the Ricci curvature $\text{Ric}_{X}\geq-(n-1)\kappa^{2}$ implies a uniform estimate of the packing function at any fixed scale $r_{0}$ , that is

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\text{Pack}(3r_{0},r_{0})\leq\frac{v_{{\mathbb{H}}^{n}_{\kappa}}(3r_{0})}{v_{{\mathbb{H}}^{n}_{\kappa}}(r_{0})}

where $v_{{\mathbb{H}}^{n}_{\kappa}}(r)$ is the volume of a ball of radius $r$ in the $n$ -dimensional space form with constant curvature $-\kappa^{2}$ .
Also remark that every metric space admitting a cocompact action is packed (for some $P_{0},r_{0}$ ), see the proof of [Cav22a, Lemma 5.4].

The packing condition has many interesting geometric consequences for complete, geodesically complete CAT $(0)$ -spaces, as showed in [CS20], [Cav21] and [Cav22b]. The first one is a uniform estimate of the measure of balls of any radius $R$ and an upper bound on the dimension, which we summarize here.

Proposition 2.3 ([CS21, Thms. 3.1, 4.2, 4.9], [Cav22b, Lemma 3.3]).

Let $X$ be a complete, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space. Then $X$ is proper and

(i)

$\textup{Pack}(R,r)\leq P_{0}(1+P_{0})^{\frac{R}{\min\{r,r_{0}\}}-1}$ for all $0<r\leq R$ ;
(ii)

the dimension of $X$ is at most $n_{0}:=P_{0}/2$ ;
(iii)

there exist functions $v,V\colon(0,+\infty)\to(0,+\infty)$ depending only on $P_{0},r_{0}$ such that for all $x\in X$ and $R>0$ we have

(7) $v(R)\leq\mu_{X}(\overline{B}(x,R))\leq V(R);$
(iv)

The entropy of $X$ is bounded above in terms of $P_{0}$ and $r_{0}$ , namely

$\textup{Ent}(X):=\limsup_{R\to+\infty}\frac{1}{R}\log\textup{Pack}(R,1)\leq\frac{\log(1+P_{0})}{r_{0}}.$

In particular, for a geodesically complete, CAT $(0)$ space $X$ which is $(P_{0},r_{0})$ -packed, the assumptions complete and proper are interchangeable.

Also, property (i) shows that, for complete and geodesically complete CAT(0)-spaces, a packing condition at some scale $r_{0}$ yields an explicit, uniform control of the packing function at any other scale $r$ : therefore, for these spaces, this condition is equivalent to similar conditions which have been considered by other authors with different names (“uniform compactness of the family of $r$ -balls” in [Gro81]; “geometrical boundedness” in [DY05], etc.).

The following remarkable version of the Margulis’ Lemma, due to Breuillard-Green-Tao, is another important consequence of a packing condition at some fixed scale. It clarifies the structure of the groups $\overline{\Gamma}_{r}(x)$ for small $r$ , which are sometimes called the “almost stabilizers”. We decline it for geodesically complete CAT $(0)$ -spaces.

Proposition 2.4 ([BGT11, Corollary 11.17]).

Given $P_{0},r_{0}>0$ , there exists $\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})>0$ such that the following holds. Let $X$ be a proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space and let $\Gamma$ be a discrete subgroup of $\textup{Isom}(X)$ : then, for every $x\in X$ and every $0\leq\varepsilon\leq\varepsilon_{0}$ , the almost stabilizer $\overline{\Gamma}_{\varepsilon}(x)$ is virtually nilpotent.

We will often refer to the constant $\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})$ as the Margulis’ constant. The conclusion of Proposition 2.4 can be improved for cocompact groups, as in this case the group $\overline{\Gamma}_{\varepsilon}(x)$ is virtually abelian (cp. [BH13, Theorem II.7.8]; indeed, a cocompact group of a CAT $(0)$ -space is always semisimple).

2.3. Crystallographic groups in the Euclidean space

We will denote points in $\mathbb{R}^{k}$ by a bold letter v, and the origin by $\bf O$ .
Among CAT $(0)$ -spaces, the Euclidean space ${\mathbb{R}}^{k}$ and its discrete groups play a special role. A crystallographic group is a discrete, cocompact group $G$ of isometries of some $\mathbb{R}^{k}$ . The simplest and most important of them, in view of Bieberbach’s Theorem, are Euclidean lattices: i.e. free abelian crystallographic groups. It is well known that a lattice must act by translations on $\mathbb{R}^{k}$ (see for instance [Far81]); so, alternatively, a lattice $\mathcal{L}$ can be seen as the set of linear combinations with integer coefficients of $k$ independent vectors $\bf b_{1},\ldots,\bf b_{k}$ (we will make no difference between a lattice and this representation). The integer $k$ is also called the rank of the lattice.
A basis $\mathcal{B}=\{\bf b_{1},\ldots,\bf b_{k}\}$ of a lattice $\mathcal{L}$ is a set of $k$ independent vectors that generate $\mathcal{L}$ as a group. There are many geometric invariants classically associated to a lattice $\mathcal{L}$ , we will need just two of them:
– the covering radius, which is defined as

\rho(\mathcal{L})=\inf\left\{r>0\text{ s.t. }\bigcup_{\bf v\in\mathcal{L}}\overline{B}({\bf v},r)=\mathbb{R}^{k}\right\}

– the shortest generating radius, that is

\lambda(\mathcal{L})=\inf\{r>0\text{ s.t. }\mathcal{L}\text{ contains }k\text{ independent vectors of length}\leq r\}.

Notice that, by the triangle inequality, any lattice $\mathcal{L}$ is $2\rho(\mathcal{L})$ -cocompact. By definition it is always possible to find a basis $\mathcal{B}=\{\bf b_{1},\ldots,\bf b_{k}\}$ of $\mathcal{L}$ such that $\|\bf b_{1}\|\leq\cdots\leq\|b_{k}\|=\lambda(\mathcal{L})$ ; this is called a shortest basis of $\mathcal{L}$ . The shortest generating radius and the covering radius are related as follows:

(8)

\rho(\mathcal{L})\leq\frac{\sqrt{k}}{2}\cdot\lambda(\mathcal{L}).

For our purposes, the content of the famous Bieberbach’s Theorems can be stated as follows.

Proposition 2.5 (Bieberbach’s Theorem).

There exists $J(k)$ , only depending on $k$ , such that the following holds true. For every crystallographic group $G$ of $\mathbb{R}^{k}$ the subgroup ${\mathcal{L}}(G)=G\cap\textup{Transl}({\mathbb{R}}^{k})$ is a normal subgroup of index at most $J(k)$ , in particular a lattice.

Here $\textup{Transl}({\mathbb{R}}^{k})$ denotes the normal subgroup of translations of ${\mathcal{E}}_{k}=\textup{Isom}(\mathbb{R}^{k})$ . The subgroup $\mathcal{L}(G)$ is called the maximal lattice of $G$ . It is well-known that every lattice ${\mathcal{L}}<G$ of rank $k$ has finite index in $G$ .

2.4. Virtually abelian groups

Recall that the (abelian, or Prüfer) rank of an abelian group $A$ , denoted $\text{rk}(A)$ , is the maximal cardinality of a subset $S\subset A$ of $\mathbb{Z}$ -linear independent elements. We extend this definition to virtually abelian groups $G$ , defining rk $(G)$ as the rank of every free abelian subgroup $A$ of finite index in $G$ : notice that if $A^{\prime}$ is a finite index subgroup of an abelian group $A$ , then $A$ and $A^{\prime}$ have same rank, so rk $(G)$ is well defined. One can equivalenty define rk $(G)$ as the rank of every normal, free abelian subgroup $A$ of finite index in $G$ , since every finite index subgroup of $G$ contains a normal, finite index subgroup. It is easy to show that the abelian rank is monotone on subgroups.

If $A$ is a discrete, finitely generated, semisimple free abelian group of isometries of a CAT $(0)$ -space $X$ , then its minimal set

\text{Min}(A):=\bigcap_{a\in A}\text{Min}(a)

is not empty and splits isometrically as $Z\times\mathbb{R}^{k}$ where $k=\textup{rk}(A)$ . This is the main content of the Flat Torus Theorem (see [BH13, Theorem II.7.1]).
We recall some additional facts about the identification $\text{Min}(A)=Z\times\mathbb{R}^{k}$ , which we will freely use later:

(a) the abelian group $A$ acts as the identity on the factor $Z$ , and cocompactly by translations on the Euclidean factor $\mathbb{R}^{k}$ ;

(b) writing $x=(z,v)\in\text{Min}(A)$ , the slice $\{z\}\times\mathbb{R}^{k}$ coincides with the convex closure $\textup{Conv}(Ax)$ of the orbit $Ax$ ;

(c) one has $\textup{Conv}(A^{\prime}x)=\textup{Conv}(Ax)$ for every finite index subgroup $A^{\prime}<A$ and every $x\in\textup{Min}(A)$ .

The last assertion follows from the fact that $\textup{Conv}(A^{\prime}x)\subseteq\textup{Conv}(Ax)$ and are both isometric to $\mathbb{R}^{k}$ , so they necessarily coincide.

As a direct consequence of the Flat Torus Theorem we have the following property for virtually abelian isometry groups of CAT $(0)$ -spaces, that we will often use.

Lemma 2.6.

Let $X$ be a proper $\textup{CAT}(0)$ -space and let $G_{0}<G$ be discrete, semisimple, virtually abelian groups of isometries of $X$ . Then $[G:G_{0}]$ is finite if and only if $G$ and $G_{0}$ have same rank.

Proof.

The implication $[G:G_{0}]<\infty\Rightarrow\text{rk}(G)=\text{rk}(G_{0})$ is trivial, as every free abelian finite index subgroup $A<G_{0}$ is also a finite index subgroup of $G$ . To show the converse implication, assume that $\text{rk}(G_{0})=\text{rk}(G)=k$ , and let $A$ be a rank $k$ , free abelian, finite index normal subgroup of $G$ . Consider the (free abelian) subgroup $A_{0}=A\cap G_{0}$ of $G_{0}$ , and notice that we have $\text{rk}(A_{0})=\text{rk}(G_{0})=k$ , since also $[G_{0}:A_{0}]=[G:A]<\infty$ . Now, both $A_{0}$ and $A$ act faithfully on the Euclidean factor of $\text{Min}(A)=Z\times{\mathbb{R}}^{k}$ (they do not contain elliptics since they are free, and act as the identity on $Z$ ). Therefore their projections $p(A_{0})<p(A)$ on $\text{Isom}({\mathbb{R}}^{k})$ are both rank $k$ Euclidean lattices, hence $[A:A_{0}]=[p(A):p(A_{0})]<\infty$ . But then we deduce that $[G:G_{0}]\leq[G:A_{0}]=[G:A][A:A_{0}]<\infty$ . ∎

The following generalization of the Flat Torus Theorem is classical.

Proposition 2.7 ([BH13, Corollary II.7.2]).

Let $X$ be a proper $\textup{CAT}(0)$ -space, and let $G$ be a discrete, semisimple, virtually abelian group of isometries of $X$ of rank $k$ . Then, there exists a closed, convex, $G$ -invariant subset $C(G)$ of $X$ which splits as $Z\times\mathbb{R}^{k}$ , satisfying the following properties:

(i)

every $g\in G$ preserves the product decomposition and acts as the identity on the first component;
(ii)

the image $G_{\mathbb{R}^{k}}$ of $G$ under the projection $G\rightarrow\textup{Isom}(\mathbb{R}^{k})$ is a crystallographic group.

Given a generating set $S$ of $G$ , the following statement explains how to construct a generating set for the maximal lattice ${\mathcal{L}}(G_{\mathbb{R}^{k}})$ with words on $S$ of bounded length. This will be used later in the proof of Theorem 4.1:

Lemma 2.8.

Same assumptions as in Proposition 2.7 above.
If $S$ is a symmetric, finite generating set for $G$ containing the identity, then there exists a subset $\Sigma\subseteq S^{4J(k)+2}$ whose projection $\Sigma_{\mathbb{R}^{k}}$ on $\textup{Isom}(\mathbb{R}^{k})$ generates $\mathcal{L}(G_{\mathbb{R}^{k}})$ , where $J(k)$ is the constant of Proposition 2.5.

Here $S^{n}\subset G$ denotes the subset of all products of at most $n$ elements of $S$ ; by definition, this coincides with $G\cap\overline{B}(e,n)$ , where $\overline{B}(e,n)$ is the closed ball of radius $n$ , centered at the identity $e$ , in the Cayley graph $\text{Cay}(G,S)$ .

Proof of Lemma 2.8.

Call $p\colon G\to G_{\mathbb{R}^{k}}$ the projection on Isom $(\mathbb{R}^{k})$ . By Proposition 2.5 the normal subgroup $\mathcal{L}(G_{\mathbb{R}^{k}})$ has index at most $J(k)$ in $G_{\mathbb{R}^{k}}$ , so the normal subgroup $G^{\prime}:=p^{-1}(\mathcal{L}(G_{\mathbb{R}^{k}}))$ has index at most $J(k)$ in $G$ . The group $G$ acts discretely by isometries on Cay $(G,S)$ with codiameter $1$ . So, by Lemma 2.1 the group $G^{\prime}$ acts on Cay $(G,S)$ with codiameter at most $2J(k)+1$ . As recalled before Lemma 2.1, $G^{\prime}$ is therefore generated by the subset $\Sigma:=G^{\prime}\cap S^{4J(k)+2}$ , made of the elements of $G^{\prime}$ displacing the point $e$ in the Cayley graph Cay $(G,S)$ at most by $4J(k)+2$ . It follows that the projection $\Sigma_{\mathbb{R}^{k}}=p(\Sigma)$ generates $\mathcal{L}(G_{\mathbb{R}^{k}})$ . ∎

Finally, remark that given a discrete, semisimple, virtually abelian group $G$ of isometries of a proper CAT $(0)$ -space, there can be several closed, convex, $G$ -invariant subsets $C(G)$ satisfying the conclusions of Proposition 2.7. What is uniquely associated to $G$ is a subset in the boundary of $X$ .

Proposition 2.9.

Same assumptions as in Proposition 2.7.
Then there exists a closed, convex, $G$ -invariant subset of $\partial X$ , denoted $\partial G$ , which is isometric to $\mathbb{S}^{k-1}$ and has the following properties:

(i)

$\partial G=\partial\textup{Conv}(Ax)$ , for every free abelian finite index subgroup $A$ of $G$ and every $x\in\textup{Min}(A)$ ;
(ii)

for every subgroup $G^{\prime}<G$ we have $\partial G^{\prime}\subseteq\partial G$ ; moreover $\partial G^{\prime}=\partial G$ if $\textup{rk}(G^{\prime})=\textup{rk}(G)$ .

The closed subset $\partial G$ provided by this proposition will be called the trace at infinity of the virtually abelian group $G$ .

Proof.

We fix a free abelian, normal subgroup $A_{0}\triangleleft G$ with finite index and $x_{0}\in\text{Min}(A_{0})$ . By the Flat Torus Theorem, $\text{Conv}(A_{0}x_{0})$ is isometric to $\mathbb{R}^{k}$ . We set $\partial G:=\partial\textup{Conv}(A_{0}x_{0})$ . Clearly $\partial G$ is closed, convex, isometric to $\mathbb{S}^{k-1}$ . Notice that the set $\partial G$ does not depend on the choice of $x_{0}\in\text{Min}(A_{0})$ , since for $x\in\text{Min}(A_{0})$ the subsets $\textup{Conv}(A_{0}x)$ and $\textup{Conv}(A_{0}x_{0})$ can be identified, by the Flat Torus Theorem, to two parallel slices $\{z\}\times\mathbb{R}^{k}$ and $\{z_{0}\}\times\mathbb{R}^{k}$ , which have the same boundary. Also, the subset $\partial G$ is $G$ -invariant: in fact, $A_{0}$ is normal in $G$ , so $\text{Min}(A_{0})$ is $G$ -invariant, therefore

g\cdot\partial G=\partial\left(g\cdot\text{Conv}(A_{0}x_{0})\right)=\partial\text{Conv}((gA_{0}g^{-1})gx_{0})=\partial\text{Conv}(A_{0}gx_{0})=\partial G

because $gx_{0}\in\text{Min}(A_{0})$ . Again by the Flat Torus Theorem (namely, property (c) recalled before), if $A<G$ is another free abelian subgroup of finite index and $x\in\text{Min}(A)$ then $\text{Conv}(Ax)=\text{Conv}((A\cap A_{0})x)=\text{Conv}((A\cap A_{0})x_{0})=\text{Conv}(A_{0}x_{0})$ , so we have $\partial\text{Conv}(Ax)=\partial G$ too. This proves (i).
To show (ii), it is enough to consider free abelian subgroups $A^{\prime}<G^{\prime}$ and $A<G$ of finite index. We can even suppose $A^{\prime}<A$ up to replacing $A^{\prime}$ by $A^{\prime}\cap A$ . If $x\in\text{Min}(A)$ then $x\in\text{Min}(A^{\prime})$ , and by the first part of the statement we have $\partial G^{\prime}=\partial\text{Conv}(A^{\prime}x)\subseteq\partial\text{Conv}(Ax)=\partial G.$ If moreover $G$ and $G^{\prime}$ have the same rank then $A^{\prime}$ is a finite index subgroup of $G$ by Lemma 2.6, and $\partial G=\partial\text{Conv}(A^{\prime}x)=\partial G^{\prime}$ . ∎

3. Systole and diastole

In this section we compare some different invariants of an action of group $\Gamma$ on a CAT $(0)$ -space $X$ , which are related to the problem of collapsing: the systole and the diastole of the action (and their corresponding free analogues), which play the role of the injectivity radius.
Recall that the systole and the free-systole of $\Gamma$ at a point $x\in X$ are defined respectively as

\text{sys}(\Gamma,x):=\inf_{g\in\Gamma^{*}}d(x,gx),\qquad\text{sys}^{\diamond}(\Gamma,x):=\inf_{g\in\Gamma^{\ast}\setminus\Gamma^{\diamond}}d(x,gx),

where $\Gamma^{*}=\Gamma\setminus\{\text{id}\}$ and $\Gamma^{\diamond}$ is the subset of all elliptic isometries of $\Gamma$ .
The (global) systole and the free-systole of $\Gamma$ are accordingly defined as

\text{sys}(\Gamma,X)=\inf_{x\in X}\text{sys}(\Gamma,x),\qquad\text{sys}^{\diamond}(\Gamma,X)=\inf_{x\in X}\text{sys}^{\diamond}(\Gamma,x).

Similarly, the diastole and the free-diastole of $\Gamma$ are defined as

\text{dias}(\Gamma,X)=\sup_{x\in X}\text{sys}(\Gamma,x),\qquad\text{dias}^{\diamond}(\Gamma,X)=\sup_{x\in X}\text{sys}^{\diamond}(\Gamma,x).

In [CS22] the authors showed that dias $(\Gamma,X)>0$ if and only if there exists a fundamental domain for $\Gamma$ ; that is, if and only if there exists a point $x_{0}$ such that the pointwise stabilizer $\text{Stab}_{\Gamma}(x_{0})$ is trivial. The actions satisfying this property will be called nonsingular, and singular when dias $(\Gamma,X)=0$ (with abuse of language, we will often say that the group $\Gamma$ itself is singular or nonsingular). For a nonsingular action, the Dirichlet domain at $x_{0}$ is defined as

\mathrm{D}_{x_{0}}=\{y\in X\text{ s.t. }d(x_{0},y)<d(x_{0},gy)\text{ for all }g\in\Gamma^{*}\}

and is always a fundamental domain for the action of $\Gamma$ , see [CS22, Prop. 2.9]. Notice that a fundamental domain exists when, for instance, $\Gamma$ is torsion-free, or when $X$ is a homology manifold, as follows by the combination of Lemma 2.1 and Proposition 2.9 of [CS22].

By definition, we have the trivial inequalities:

\text{sys}(\Gamma,X)\leq\text{sys}^{\diamond}(\Gamma,X)\leq\text{dias}^{\diamond}(\Gamma,X)

\text{sys}(\Gamma,X)\leq\text{dias}(\Gamma,X)\leq\text{dias}^{\diamond}(\Gamma,X).

The following result shows that the free systole and the free diastole are for small values quantitatively equivalent, provided one knows an a priori bound on the diameter of the quotient. Moreover, for nonsingular actions, both are quantitatively equivalent to the diastole.

Theorem 3.1.

Given $P_{0},r_{0},D_{0}$ , there exists $b_{0}=b_{0}(P_{0},r_{0})>0$ such that the following holds true. Let $X$ be a proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space and let $\Gamma<\textup{Isom}(X)$ be discrete and $D_{0}$ -cocompact. Then:

(i)

$\textup{sys}^{\diamond}(\Gamma,X)\geq\left(1+P_{0}\right)^{-\frac{(2D_{0}+1)}{\min\{\textup{dias}^{\diamond}(\Gamma,X),r_{0}\}}}$ .
(ii)

If moreover the action of $\Gamma$ on $X$ is nonsingular then

$\textup{dias}(\Gamma,X)\geq b_{0}\cdot\min\{\textup{sys}^{\diamond}(\Gamma,X),\varepsilon_{0}\}.$

(Here, $\varepsilon_{0}$ is the Margulis’ constant given by Proposition 2.4).

Dropping the assumption of nonsingularity. (ii) is no longer true: see [CS22, Example 1.4], where dias $(\Gamma,X)=0$ while the free systole is positive. Also, it is easy to convince oneself that the usual systole is not equivalent, for small values, to the other three invariants: for instance, for every discrete, cocompact action of a group $\Gamma$ on a proper CAT $(0)$ -space $X$ , one has $\text{dias}(\Gamma,X)>0$ if there exists some point trivial stabilizer, but clearly $\text{sys}(\Gamma,X)=0$ if $\Gamma$ has torsion. Finally, notice that the inequality (ii) holds for a constant depending only on $P_{0}$ and $r_{0}$ , and not on $D_{0}$ ; it is not difficult to show that the same is not true for (i).

To show the above equivalences, we need two auxiliary facts.
The first one is a generalization of Buyalo’s and Cao-Cheeger-Rong’s theorem about the existence of abelian, local splitting structure, for groups acting faithfully and geometrically on packed, CAT $(0)$ -spaces which are $\varepsilon$ -thin for sufficiently small $\varepsilon$ .

Proposition 3.2 ([CS22, Theorem A & Remark 3.5]).

Let $P_{0},r_{0}>0$ and fix $0<\lambda\leq\varepsilon_{0}$ , where $\varepsilon_{0}$ is given by Proposition 2.4. There exists a constant $b_{0}=b_{0}(P_{0},r_{0})>0$ such that the following holds true. Let $X$ be a proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space and let $\Gamma<\textup{Isom}(X)$ be discrete and cocompact. If dias $(\Gamma,X)\leq b_{0}\cdot\lambda$ then

X=\bigcup_{g\in\Gamma^{*},\,\ell(g)\leq\lambda}\textup{Min}(g).

The second fact is the following result, which propagates the smallness of the systole for torsion-free cyclic groups from point to point.

Proposition 3.3.

Let $P_{0},r_{0},R>0$ and $0<\varepsilon\leq r_{0}$ . Then there exists $\delta(P_{0},r_{0},R,\varepsilon)>0$ with the following property. Let $X$ be a proper, geodesically complete, $(P_{0},r_{0})$ -packed, CAT $(0)$ -space. If $g$ is an isometry of $X$ with infinite order and $x$ is a point of $X$ such that $d(x,gx)\leq\delta(P_{0},r_{0},R,\varepsilon)$ , then for every $y\in X$ with $d(x,y)\leq R$ there exists $m\in\mathbb{Z}^{*}$ such that $d(y,g^{m}y)\leq\varepsilon$ .
(We can choose, explicitely, $\delta(P_{0},r_{0},R,\varepsilon)=(1+P_{0})e^{-\frac{(2R+1)}{\varepsilon}}$ ).

Proof.

This follows immediately from [CS20, Proposition 4.5.(ii)], where we expressed this property in terms of the distance between generalized Margulis domains²²2Notice that in [CS20] we were in the setting of Gromov-hyperbolic spaces with a geodesically complete convex geodesic bicombing (in particular, Gromov-hyperbolic, geodesically complete CAT $(0)$ -spaces); but in that proof we never used the hyperbolicity.. ∎

Proof of Theorem 3.1.

Assume that $\min\{\text{dias}^{\diamond}(\Gamma,X),r_{0}\}>\varepsilon$ . By definition there exists $x_{0}\in X$ such that for every hyperbolic isometry $g\in\Gamma$ one has $d(x_{0},gx_{0})>\varepsilon$ . Now, if sys ${}^{\diamond}(\Gamma,X)\leq(1+P_{0})e^{-\frac{(2D_{0}+1)}{\varepsilon}}=:\delta$ , we could find $x\in X$ and a hyperbolic $g\in\Gamma$ such that $d(x,gx)\leq\delta$ . By Proposition 3.3, for every $y\in B(x,D_{0})$ there would exists a non trivial power $g^{m}$ satisfying $d(y,g^{m}y)\leq\varepsilon$ . But then, since the action is $D_{0}$ -cocompact we could find a conjugate $\gamma$ of $g^{m}$ (thus, a hyperbolic isometry) such that $d(x_{0},\gamma x_{0})\leq\varepsilon$ , a contradiction. The conclusion follows by the arbitrariness of $\varepsilon$ .
To see (ii), take any $\lambda<\min\{\text{sys}^{\diamond}(\Gamma,X),\varepsilon_{0}\}$ . If $\text{dias}(\Gamma,X)\leq b_{0}\cdot\lambda$ , where $b_{0}$ is the constant given by Proposition 3.2, we conclude that $X=\bigcup_{g}\text{Min}(g)$ where $g$ runs over all nontrivial elements with $\ell(g)\leq\lambda$ . But by definition $\ell(g)>\lambda$ for all hyperbolic $g$ , so $X=\bigcup_{g\in\Gamma^{\diamond}}\text{Min}(g).$ Hence, $\text{dias}(\Gamma,X)=0$ , contradicting the nonsingularity of $\Gamma$ . As $\lambda$ is arbitrary, this proves (ii). ∎

4. The splitting theorem

In this section we will prove Theorem D, actually a stronger, parametric version given by Theorem 4.1 below. To set the notation, recall that for a proper, geodesically complete, CAT $(0)$ -space $X$ which is $(P_{0},r_{0})$ -packed we have an upper bound on the dimension of $X$ given by Proposition 2.3

\dim(X)\leq n_{0}=P_{0}/2

and a Margulis’s constant $\varepsilon_{0}$ (only depending on $P_{0},r_{0}$ ) given by Proposition 2.4. Also, recall the constant $J(k)$ given by Proposition 2.5, and define

J_{0}:=\max_{k\in\{0,\ldots,n_{0}\}}J(k)+1

which also clearly depends only on $n_{0}$ , so ultimately only on $P_{0}$ .
Finally, for a discrete subgroup $\Gamma<\textup{Isom}(X)$ recall the definition (4) of the subgroup $\overline{\Gamma}_{r}(x)<\Gamma$ generated by $\overline{\Sigma}_{r}(x)$ given in Section 2.1.

Theorem 4.1.

Given positive constants $P_{0},r_{0},D_{0}$ , there exists a function $\sigma_{P_{0},r_{0},D_{0}}:(0,\varepsilon_{0}]\rightarrow(0,\varepsilon_{0}]$ (depending only on the parameters $P_{0},r_{0},D_{0}$ ) such that the following holds. Let $X$ be a proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -space, and $\Gamma<\textup{Isom}(X)$ be discrete and $D_{0}$ -cocompact. For every chosen $\varepsilon\in(0,\varepsilon_{0}]$ , if $\textup{sys}^{\diamond}(\Gamma,X)\leq\sigma_{P_{0},r_{0},D_{0}}(\varepsilon)$ then:

(i)

the space $X$ splits isometrically as $Y\times\mathbb{R}^{k}$ , with $k\geq 1$ , and this splitting is $\Gamma$ -invariant;
(ii)

there exists ${\varepsilon^{\ast}}\in(\sigma_{P_{0},r_{0},D_{0}}(\varepsilon),\varepsilon)$ such that the rank of the virtually abelian subgroups $\overline{\Gamma}_{{\varepsilon^{\ast}}}(x)$ is exactly $k$ , for all $x\in X$ ;
(iii)

the traces at infinity $\partial\overline{\Gamma}_{{\varepsilon^{\ast}}}(x)$ equal the boundary ${\mathbb{S}}^{k-1}$ of the convex subsets $\{y\}\times\mathbb{R}^{k}$ , for all $x\in X$ and all $y\in Y$ ;
(iv)

for every $x\in X$ there exists $y\in Y$ such that $\overline{\Gamma}_{\varepsilon^{\ast}}(x)$ preserves $\{y\}\times\mathbb{R}^{k}$ ;
(v)

the projection of $\overline{\Gamma}_{\varepsilon^{\ast}}(x)$ on $\textup{Isom}(\mathbb{R}^{k})$ is a crystallographic group, whose maximal lattice is generated by the projection of a subset $\Sigma\subset\Sigma_{4J_{0}\cdot{\varepsilon^{\ast}}}(x)$ ;
(vi)

the closure of the projection of $\overline{\Gamma}_{\varepsilon^{\ast}}(x)$ on $\textup{Isom}(Y)$ is compact and totally disconnected.

Here, by $\Gamma$ -invariant splitting we mean that every isometry of $\Gamma$ preserves the product decomposition. By of [BH13, Proposition I.5.3.(4)] we can see $\Gamma$ as a subgroup of $\textup{Isom}(Y)\times\textup{Isom}(\mathbb{R}^{k})$ . In particular it is meaningful to talk about the projection of $A$ on $\text{Isom}(Y)$ and $\text{Isom}(\mathbb{R}^{k})$ .

Observe that Theorem D is a special case of Theorem 4.1.(i), for $\varepsilon=\varepsilon_{0}$ , which yields the constant $\sigma_{0}=\sigma_{P_{0},r_{0},D_{0}}(\varepsilon_{0})$ . We call the integer $1\leq k\leq n_{0}$ given by (ii) the ${\varepsilon^{\ast}}$ -splitting rank of $X$ .

To prove Theorem 4.1 we need a little of preparation.
The first step will be to exhibit a free abelian subgroup of rank $k\geq 1$ which is commensurated in $\Gamma$ . Recall that two subgroups $G_{1},G_{2}<\Gamma$ are called commensurable in $\Gamma$ if the intersection $G_{1}\cap G_{2}$ has finite index in both $G_{1}$ and $G_{2}$ . A subgroup $G<\Gamma$ is said to be commensurated in $\Gamma$ if $G$ and $\gamma G\gamma^{-1}$ are commensurable in $\Gamma$ for every $\gamma\in\Gamma$ .

Now, we fix $0<\varepsilon\leq\varepsilon_{0}$ as in the assumptions of Theorem 4.1, and we define inductively the sequence of subgroups $\overline{\Gamma}_{\varepsilon_{i}}(x)$ associated to positive numbers

\varepsilon_{1}:=\varepsilon>\varepsilon_{2}>\ldots>\varepsilon_{2n_{0}+1}>0

as follows:
– first, we apply Proposition 3.3 to $\varepsilon=\varepsilon_{1}$ and $R=2D_{0}$ to obtain a smaller $\delta_{2}:=\delta(P_{0},r_{0},2D_{0},\varepsilon_{1})$ , and we set $\varepsilon_{2}:=\delta_{2}/4J_{0}$ ;
– then, we define inductively $\delta_{i+1}:=\delta(P_{0},r_{0},2D_{0},\varepsilon_{i})$ by repeatedly applying Proposition 3.3 to $\varepsilon_{i}$ and $R=2D_{0}$ , and we set $\varepsilon_{i+1}=\delta_{i+1}/4J_{0}$ .
Notice that, by construction, each $\varepsilon_{i}$ depends only on $P_{0},r_{0},D_{0}$ and $\varepsilon$ .
By Proposition 2.4, the subgroups $\overline{\Gamma}_{\varepsilon_{i}}(x)$ form a decreasing sequence of virtually abelian, semisimple subgroups for every $x\in X$ .

We set $\sigma_{P_{0},r_{0},D_{0}}(\varepsilon):=\varepsilon_{2n_{0}+1}$ and we will show that this is the function of $\varepsilon$ for which Theorem 4.1 holds; it clearly depends only on $P_{0},r_{0},D_{0}$ and $\varepsilon$ . In what follows, we will write for short $\sigma:=\sigma_{P_{0},r_{0},D_{0}}(\varepsilon)$ .

Lemma 4.2.

If $\textup{rk}(\overline{\Gamma}_{\sigma}(x))\geq 1$ then there exists $i\in\{2,\ldots,2n_{0}\}$ such that $\textup{rk}(\overline{\Gamma}_{\varepsilon_{i+1}}(x))=\textup{rk}(\overline{\Gamma}_{\varepsilon_{i}}(x))=\textup{rk}(\overline{\Gamma}_{\varepsilon_{i-1}}(x))\geq 1$ .

Proof.

The subgroups $\overline{\Gamma}_{\varepsilon_{i}}(x)$ are virtually abelian with $\textup{rk}(\overline{\Gamma}_{\varepsilon_{i}}(x))\geq 1$ , for all $1\leq i\leq 2n_{0}+1$ , since they contain $\overline{\Gamma}_{\sigma}(x)$ . Moreover, by Proposition 2.7, the rank of each $\overline{\Gamma}_{\varepsilon_{i}}(x)$ cannot exceed the dimension of $X$ , which is at most $n_{0}$ . Since the rank decreases as $i$ increases we conclude that for some $2\leq i\leq 2n_{0}$ we have $\textup{rk}(\overline{\Gamma}_{\varepsilon_{i+1}}(x))=\textup{rk}(\overline{\Gamma}_{\varepsilon_{i}}(x))=\textup{rk}(\overline{\Gamma}_{\varepsilon_{i-1}}(x))\geq 1$ . ∎

Proposition 4.3.

If $\textup{rk}(\overline{\Gamma}_{\sigma}(x))\geq 1$ then there exists a free abelian subgroup $A<\overline{\Gamma}_{\varepsilon_{i}}(x)$ of rank $k\geq 1$ , which has finite index in $\overline{\Gamma}_{\varepsilon_{i-1}}(x)$ and is commensurated in $\Gamma$ , for some $i\in\{2,\ldots,2n_{0}\}$ .

Proof.

Consider the virtually abelian groups $\overline{\Gamma}_{\varepsilon_{i+1}}(x)<\overline{\Gamma}_{\varepsilon_{i}}(x)<\overline{\Gamma}_{\varepsilon_{i-1}}(x)$ which have the same rank $k\geq 1$ , given by Lemma 4.2, for some $2\leq i\leq 2n_{0}$ and let $A_{\varepsilon_{j}}(x)<\overline{\Gamma}_{\varepsilon_{j}}(x)$ be free abelian, finite index subgroups of rank $k$ , for $j=i-1,i,i+1$ . Notice that $A_{\varepsilon_{i+1}}(x)$ has finite index also in $\overline{\Gamma}_{\varepsilon_{i}}(x)$ , by Lemma 2.6. Then, let $C_{i+1}=Z_{i+1}\times{\mathbb{R}}^{k}$ be the $\overline{\Gamma}_{\varepsilon_{i+1}}(x)$ -invariant, convex subset of $X$ given by Proposition 2.7, applied to $\overline{\Gamma}_{\varepsilon_{i+1}}(x)$ , and call $\bar{\Gamma}_{i+1}$ the image of $\overline{\Gamma}_{\varepsilon_{i+1}}(x)$ under the projection $p_{i+1}:\overline{\Gamma}_{\varepsilon_{i+1}}(x)\rightarrow\text{Isom}(\mathbb{R}^{k})$ on the second factor of $C_{i+1}$ . Finally, denote by ${\mathcal{L}}_{i+1}$ the maximal Euclidean lattice of the crystallograhic group $\bar{\Gamma}_{i+1}$ .

By Lemma 2.8 we can find a subset $\Sigma$ of $\overline{\Sigma}_{\varepsilon_{i+1}}(x)^{4J_{0}}\subseteq\overline{\Sigma}_{4J_{0}\cdot\varepsilon_{i+1}}(x)$ whose projection $\Sigma_{\mathbb{R}^{k}}=p_{i+1}(\Sigma)$ on Isom $(\mathbb{R}^{k})$ generates the lattice ${\mathcal{L}}_{i+1}$ . In particular every non-trivial element of $\Sigma$ is hyperbolic.
Moreover, by the definition of $\varepsilon_{i+1}=\delta_{i+1}/4J_{0}$ , the following holds:

$\forall g\in\Sigma$ and $\forall h\in\overline{\Sigma}_{2D_{0}}(x)$ there exists $m>0$ such that $h^{-1}g^{m}h\in\overline{\Sigma}_{\varepsilon_{i}}(x)$

(in fact, $d(x,gx)\leq\delta_{i+1}=\delta(P_{0},r_{0},2D_{0},{\varepsilon_{i}})$ for every $g\in\Sigma$ , so by Proposition 3.3 there exists $m>0$ such that $d(x,h^{-1}g^{m}hx)=d(hx,g^{m}hx)\leq\varepsilon_{i}$ , that is $h^{-1}g^{m}h\in\overline{\Sigma}_{\varepsilon_{i}}(x)$ ).
Since $\Sigma$ and $\overline{\Sigma}_{2D_{0}}(x)$ are finite sets, we can then find a positive integer $M$ such that $hg^{M}h^{-1}\in{\overline{\Gamma}_{\varepsilon_{i}}}(x)$ for all $g\in\Sigma$ and all $h\in\overline{\Sigma}_{2D_{0}}(x)$ .
Moreover, we can even choose $M>0$ so that $\forall g\in\Sigma$ and $\forall h\in\overline{\Sigma}_{2D_{0}}(x)$ the elements $g^{M}$ and $hg^{M}h^{-1}$ belong, respectively, to the free abelian, finite index subgroups $A_{\varepsilon_{i+1}}(x)$ and $A_{\varepsilon_{i}}(x)$ of $\overline{\Gamma}_{\varepsilon_{i}}(x)$ .
Now, let $A<A_{\varepsilon_{i+1}}(x)$ be the (free abelian) subgroup generated by the subset $S=\{g^{M}\,\;|\;\,g\in\Sigma\}$ . We claim that $A$ is commensurated in $\Gamma$ .
Actually, notice first that $A$ has rank equal to $k$ , since its projection $p_{i+1}(A)$ is a subgroup of finite index of the lattice ${\mathcal{L}}_{i+1}$ of ${\mathbb{R}^{k}}$ . Therefore, for all $h\in\overline{\Sigma}_{2D_{0}}(x)$ , the free abelian group $hAh^{-1}$ has also rank $k$ , and is contained in the free abelian group $A_{\varepsilon_{i}}(x)$ of same rank. This implies, again by Lemma 2.6, that $A$ and $hAh^{-1}$ have finite index in $A_{\varepsilon_{i}}(x)$ , and that $A\cap hAh^{-1}$ has finite index in both $A$ and $hAh^{-1}$ . Hence $A$ and $hAh^{-1}$ are commensurable for every $h\in\overline{\Sigma}_{2D_{0}}(x)$ .

Now, given $g\in\Gamma$ , we can write it as $g=h_{1}\cdots h_{n}$ for some $h_{i}\in\overline{\Sigma}_{2D_{0}}(x)$ and set $g_{k}:=h_{1}\cdots h_{k}$ . We clearly have

g_{k}Ag_{k}^{-1}\cap g_{k+1}Ag_{k+1}^{-1}=g_{k}(A\cap h_{k+1}Ah_{k+1}^{-1})g_{k}^{-1}

with $A\cap h_{k+1}Ah_{k+1}^{-1}$ of finite index in both factors; so $g_{k}Ag_{k}^{-1}$ and $g_{k+1}Ag_{k+1}^{-1}$ are commensurable. Since commensurability in a group is a transitive relation, this shows that $A$ and $gAg^{-1}$ are commensurable for all $g\in\Gamma$ .
Finally, observe that $A<\overline{\Gamma}_{\varepsilon_{i-1}}(x)$ and these groups have same rank, so $A$ has finite index also in $\overline{\Gamma}_{\varepsilon_{i-1}}(x)$ , again by Lemma 2.6. ∎

We need now to recall an additional notion. Given $Z\subseteq\partial X$ we say that a subset $Y\subseteq X$ is $Z$ -boundary-minimal if it is closed, convex, $\partial Y=Z$ and $Y$ is minimal with this properties. The union of all the $Z$ -boundary-minimal sets is denoted by Bd-Min $(Z)$ . A particular case of [CM09b, Proposition 3.6] reads as follows.

Lemma 4.4.

Let $X$ be a proper $\textup{CAT}(0)$ -space and let $Z$ be a closed, convex subset of $\partial X$ which is isometric to $\mathbb{S}^{k-1}$ . Then each $Z$ -boundary-minimal subset of $X$ is isometric to $\mathbb{R}^{k}$ and Bd-Min $(Z)$ is a closed, convex subset of $X$ which splits isometrically as $Y\times\mathbb{R}^{k}$ . Moreover $Z$ coincides with the boundary at infinity of all the slices $\{y\}\times\mathbb{R}^{k}$ , for $y\in Y$ .

A consequence of Lemma 4.4 for a commensurated group $A$ of $\Gamma$ is the following.

Proposition 4.5.

Same assumptions as in Theorem 4.1.
If $A$ is a free abelian, commensurated subgroup of $\Gamma$ of rank $k$ then we have $X=\textup{Bd-Min}(\partial A)$ and $X$ splits isometrically and $\Gamma$ -invariantly as $Y\times\mathbb{R}^{k}$ . Moreover, the projection of $A$ on $\textup{Isom}(\mathbb{R}^{k})$ is a lattice and the closure of the projection of $A$ on $\textup{Isom}(Y)$ is compact and totally disconnected.
The splitting $X=Y\times\mathbb{R}^{k}$ satisfies the following properties:

(i)

the trace at infinity $\partial A$ is $\Gamma$ -invariant and coincides with the boundary of each slice $\{y\}\times\mathbb{R}^{k}$ , for all $y\in Y$ ;
(ii)

if $A^{\prime}<A$ is another free abelian, commensurated subgroup of $\Gamma$ of rank $k^{\prime}$ then the splittings $X=Y\times\mathbb{R}^{k}$ and $X=Y^{\prime}\times\mathbb{R}^{k^{\prime}}$ associated respectively to $A$ and $A^{\prime}$ are compatible, i.e. $Y^{\prime}$ is isometric to $Y\times\mathbb{R}^{k-k^{\prime}}$ .

Proof.

The trace at infinity $\partial A$ of $A$ is isometric to $\mathbb{S}^{k-1}$ , by Proposition 2.9. We claim that it is $\Gamma$ -invariant. Indeed, if $g\in\Gamma$ then $A\cap gAg^{-1}$ has finite index in both $A$ and $gAg^{-1}$ . Then, the characterization of the trace at infinity given in Proposition 2.9 implies that

\partial A=\partial\left(A\cap gAg^{-1}\right)=\partial\left(gAg^{-1}\right)=g\partial A.

By Lemma 4.4 applied to $Z=\partial A$ we deduce that $\text{Bd-Min}(\partial A)$ is a closed, convex subset of $X$ which splits isometrically as $Y\times\mathbb{R}^{k}$ , and that $\partial A$ coincides with the boundary at infinity of all sets $\{y\}\times\mathbb{R}^{k}$ .
Now, each element of $\Gamma$ sends a $\partial A$ -boundary-minimal subset into a $\partial A$ -boundary-minimal subset because $\partial A$ is $\Gamma$ -invariant, therefore $\text{Bd-Min}(\partial A)$ itself is $\Gamma$ -invariant. Since $\Gamma$ is cocompact, the action of $\Gamma$ on $X$ is minimal: that is, if $C\neq\emptyset$ is a closed, convex, $\Gamma$ -invariant subset of $X$ then $C=X$ (see [CM09b, Lemma 3.13]). Therefore we deduce that $X=\text{Bd-Min}(\partial A)$ , and so $X$ splits isometrically and $\Gamma$ -invariantly as $Y\times\mathbb{R}^{k}$ , which proves the first assertion and (i).
The fact that the projection of $A$ on $\textup{Isom}(\mathbb{R}^{k})$ is a lattice follows from the Flat Torus Theorem. To study the projection of $A$ on $\text{Isom}(Y)$ , recall that by Proposition 2.2 we can split $X$ as $M\times\mathbb{R}^{n}\times N$ , for some $n\geq k$ , and $\textup{Isom}(X)$ as ${\mathcal{S}}\times{\mathcal{E}}_{n}\times{\mathcal{D}}$ , where ${\mathcal{S}}$ is a semi-simple Lie group with trivial center and without compact factors, ${\mathcal{E}}_{n}\cong\text{Isom}(\mathbb{R}^{n})$ and ${\mathcal{D}}$ is totally disconnected. Therefore $Y=M\times\mathbb{R}^{n-k}\times N$ and $\textup{Isom}(Y)$ splits as ${\mathcal{S}}\times{\mathcal{E}}_{n-k}\times{\mathcal{D}}$ . Moreover, $\text{Min}(A)=Z\times\mathbb{R}^{k}$ with $Z\subset Y$ . Since $A$ acts as the identity on $Z$ , it follows that the projection of $A$ on $\textup{Isom}(Y)$ fixes some point $z\in Y$ , hence its closure is a compact group. Finally, Theorem 2.(i) of [CM19] and the beginning of the proof therein show that the projection of $A$ on ${\mathcal{S}}$ is finite and the projection of $A$ on ${\mathcal{E}}_{n}$ is discrete; as the projection of $A$ on ${\mathcal{E}}_{k}$ is a lattice, then also the projection on ${\mathcal{E}}_{n-k}$ is discrete. This, combined with the fact that ${\mathcal{D}}$ is totally disconnected, implies that the closure of the projection of $A$ on $\text{Isom}(Y)$ is totally disconnected.
Suppose now to have another abelian subgroup $A^{\prime}<A$ of rank $k^{\prime}$ which is commensurated in $\Gamma$ . Let $X=Y^{\prime}\times\mathbb{R}^{k^{\prime}}$ be the splitting associated to $A^{\prime}$ . Let $x\in X$ be a point and write it as $(y,{\bf v})\in Y\times\mathbb{R}^{k}$ and $(y^{\prime},{\bf v^{\prime}})\in Y^{\prime}\times\mathbb{R}^{k^{\prime}}$ . Then, by the first part of the proof and by Proposition 2.9 we have

\partial(\{y^{\prime}\}\times\mathbb{R}^{k^{\prime}})=\partial A^{\prime}\subseteq\partial A=\partial(\{y\}\times\mathbb{R}^{k}).

It follows that $\{y^{\prime}\}\times\mathbb{R}^{k^{\prime}}\subseteq\{y\}\times\mathbb{R}^{k}$ , so the parallel slices associated to $A^{\prime}$ are contained in the parallel slices associated to $A$ . Decomposing $\mathbb{R}^{k}$ as the orhogonal sum of $\mathbb{R}^{k^{\prime}}$ and $\mathbb{R}^{k-k^{\prime}}$ , we also deduce that the sets $\{y\}\times\mathbb{R}^{k-k^{\prime}}$ are parallel for all $y\in Y$ (since the slices of $A^{\prime}$ are all parallel). Therefore, $X$ is also isometric to $(Y\times\mathbb{R}^{k-k^{\prime}})\times\mathbb{R}^{k^{\prime}}$ , which implies that $Y^{\prime}$ is isometric to $Y\times\mathbb{R}^{k-k^{\prime}}$ and proves (ii). ∎

Putting the ingredients all together we can give the

Proof of Theorem 4.1.

We show that the statement holds for ${\varepsilon^{\ast}}=\varepsilon_{i}\in(\sigma,\varepsilon)$ , where $\varepsilon_{i}$ is given by Proposition 4.3. In fact, since $\textup{sys}^{\diamond}(\Gamma,X)\leq\sigma$ , then there exists $x_{0}\in X$ with $\textup{sys}^{\diamond}(\Gamma,x_{0})\leq\sigma$ ; in particular, $\overline{\Gamma}_{\sigma}(x_{0})$ contains a hyperbolic isometry, hence $\text{rk}(\overline{\Gamma}_{\sigma}(x_{0}))\geq 1$ . Then, we can apply Proposition 4.3 and find a free abelian, commensurated subgroup $A_{0}<\Gamma$ of rank $k\geq 1$ , with $A_{0}\subset\overline{\Gamma}_{\varepsilon_{i+1}}(x_{0})$ and with finite index in $\overline{\Gamma}_{\varepsilon_{i-1}}(x_{0})$ , for some $2\leq i\leq 2n_{0}$ ; Proposition 4.5 now implies that $X$ splits isometrically and $\Gamma$ -invariantly as $Y\times\mathbb{R}^{k}$ , proving (i). Moreover, we know that $\partial A_{0}$ coincides with the boundary at infinity of each slice $\{y\}\times\mathbb{R}^{k}$ .
Let us now study the properties (ii)-(vi) for the groups $\overline{\Gamma}_{{\varepsilon_{i}}}(x)$ .
We start proving them for every $x\in X$ such that $d(x,x_{0})\leq D_{0}$ . By Proposition 3.3 and by definition of $\varepsilon_{i}$ , for every hyperbolic $g\in\overline{\Sigma}_{\varepsilon_{i}}(x)$ there exists $m\in\mathbb{Z}^{*}$ such that $d(x_{0},g^{m}x_{0})\leq\varepsilon_{i-1}$ , that is $g^{m}\in\overline{\Gamma}_{\varepsilon_{i-1}}(x_{0})$ . Let $A_{\varepsilon_{i}}(x)<\overline{\Gamma}_{\varepsilon_{i}}(x)$ be free abelian of finite index, so $\text{rk}(A_{\varepsilon_{i}}(x))=\text{rk}\left(\overline{\Gamma}_{\varepsilon_{i}}(x)\right)$ by definition. Since the set $\overline{\Sigma}_{\varepsilon_{i}}(x)$ is finite, we can find $M\in\mathbb{Z}^{*}$ such that $g^{M}\in A_{\varepsilon_{i}}(x)\cap\overline{\Gamma}_{\varepsilon_{i-1}}(x_{0})$ for every hyperbolic $g\in\overline{\Sigma}_{\varepsilon_{i}}(x)$ . So, if we define the subgroup

A:=\langle g^{M}\,\;|\,\;g\in\overline{\Sigma}_{\varepsilon_{i}}(x)\text{ hyperbolic}\rangle

we have $A<A_{\varepsilon_{i}}(x)\cap\overline{\Gamma}_{\varepsilon_{i-1}}(x_{0})$ . Notice that $A$ is again free abelian of rank $k$ , hence it has finite index in $A_{\varepsilon_{i}}(x)$ and $\overline{\Gamma}_{\varepsilon_{i}}(x)$ , by Lemma 2.6. Thus

\text{rk}(\overline{\Gamma}_{\varepsilon_{i}}(x))=\text{rk}(A)\leq\text{rk}(\overline{\Gamma}_{\varepsilon_{i-1}}(x_{0}))=\text{rk}(A_{0})=k.

Moreover by Proposition 2.9 we have $\partial\overline{\Gamma}_{\varepsilon_{i}}(x)=\partial A\subseteq\partial\overline{\Gamma}_{\varepsilon_{i-1}}(x_{0})=\partial A_{0}$ . Reversing the roles of $x$ and $x_{0}$ and starting from $\overline{\Gamma}_{\varepsilon_{i+1}}(x_{0})$ we obtain the opposite estimate $k=\text{rk}(A_{0})=\text{rk}(\overline{\Gamma}_{\varepsilon_{i+1}}(x_{0}))\leq\text{rk}(\overline{\Gamma}_{\varepsilon_{i}}(x))$ and $\partial A_{0}\subseteq\partial\overline{\Gamma}_{\varepsilon_{i}}(x)$ , which proves (ii) and (iii) in this case.
By construction $\overline{\Gamma}_{\varepsilon_{i}}(x)\cap A$ has finite index in both $\overline{\Gamma}_{\varepsilon_{i}}(x)$ and $A$ , so also the projection $p_{\mathbb{R}^{k}}(\overline{\Gamma}_{\varepsilon_{i}}(x))$ on $\textup{Isom}(\mathbb{R}^{k})$ is discrete, and the closure of the projection $p_{Y}(\overline{\Gamma}_{\varepsilon^{\ast}}(x))$ of $\overline{\Gamma}_{\varepsilon_{i}}(x)$ on $\text{Isom}(Y)$ is compact and totally disconnected. Notice that $p_{\mathbb{R}^{k}}(\overline{\Gamma}_{\varepsilon_{i}}(x))$ is cocompact, so it is a crystallographic group of $\mathbb{R}^{k}$ ; moreover, as $\Sigma_{\varepsilon_{i}}(x)$ generates $\overline{\Gamma}_{\varepsilon_{i}}(x)$ , then the maximal lattice of $p_{\mathbb{R}^{k}}(\overline{\Gamma}_{\varepsilon_{i}}(x))$ is generated by a subset of $\Sigma_{4J_{0}\cdot\varepsilon_{i}}(x)$ , by Lemma 2.8. This proves (v) and (vi). Moreover, the precompact group $p_{Y}(\overline{\Gamma}_{\varepsilon_{i}}(x))$ has a fixed point $y\in Y$ ([BH13, Corollary II.2.8.(1)]), so $\{y\}\times\mathbb{R}^{k}$ is preserved by $\overline{\Gamma}_{\varepsilon_{i}}(x)$ , proving (iv). Finally, assume that $x^{\prime}$ is a point of $X$ , say $x^{\prime}=gx$ with $d(x,x_{0})\leq D_{0}$ . Observe that $\overline{\Gamma}_{\varepsilon_{i}}(x^{\prime})=g\overline{\Gamma}_{\varepsilon_{i}}(x)g^{-1}$ , so the rank does not change and the conditions (iv) and (v) continue to hold since the splitting is $\Gamma$ -invariant. Moreover $\partial\overline{\Gamma}_{\varepsilon_{i}}(x^{\prime})=g\cdot\partial\overline{\Gamma}_{\varepsilon_{i}}(x)=g\cdot\partial A=A$ , because $\partial A$ is $\Gamma$ -invariant, so (iii) still holds. Finally, if the group $p_{Y}(\overline{\Gamma}_{\varepsilon_{i}}(x))$ preserves $\{y\}\times\mathbb{R}^{k}$ , then $p_{Y}(\overline{\Gamma}_{\varepsilon_{i}}(x^{\prime}))$ clearly preserves $\{gy\}\times\mathbb{R}^{k}$ , which proves (iv). ∎

5. The finiteness Theorems

The goal of this section is to prove Theorem A and its corollaries. The work is divided into two steps: we first prove the renormalization Theorem E, from which we immediately deduce the finiteness up to group isomorphism, and Corollary F. Then, we will improve the result showing the finiteness of the class ${\mathcal{O}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$ (and of ${\mathcal{L}}\text{-CAT}_{0}(P_{0},r_{0},D_{0})$ as a particular case) up to equivariant homotopy equivalence of orbispaces, that is Corollary B). Finally, we will deduce Corollary C from the renormalization Theorem E combined with Cheeger’s and Fukaya’s finiteness theorems.

5.1. Finiteness up to group isomorphism

Recall that a marked group is a group $\Gamma$ endowed with a generating set $\Sigma$ . Two marked groups $(\Gamma,\Sigma)$ and $(\Gamma^{\prime},\Sigma^{\prime})$ are equivalent if there exists a group isomorphism $\phi:\Gamma\rightarrow\Gamma^{\prime}$ such that $\phi(\Sigma)=\Sigma^{\prime}$ ; notice that such a $\phi$ induces an isometry between the respective Cayley graphs.
The first step for Theorem A is the following:

Proposition 5.1.

Let $P_{0},r_{0},D_{0}$ be given. For every fixed $D\geq D_{0}$ and $s>0$ , there exist only finitely many marked groups $(\Gamma,\Sigma)$ where:

–

$\Gamma$ is a discrete, $D_{0}$ -cocompact isometry group of a proper, geodesically complete, $(P_{0},r_{0})$ -packed, CAT $(0)$ -space $X$ satisfying $\textup{sys}(\Gamma,x)\geq s$ for some $x\in X$ ,
–

$\Sigma=\overline{\Sigma}_{2D}(x)$ is a $2D$ -short generating set of $\Gamma$ at $x$ .

(Finiteness here is meant up to equivalence of marked groups.)

Proof.

Recall from Section 2.1 that for any $D\geq D_{0}$ the group $\Gamma$ admits a presentation as $\Gamma=\langle\overline{\Sigma}_{2D}(x)\hskip 2.84526pt|\hskip 2.84526pt{\mathcal{R}}_{2D}(x)\rangle$ , where ${\mathcal{R}}_{2D}(x)$ is a subset of words of length $3$ on the alphabet $\overline{\Sigma}_{2D}(x)$ . Therefore the number of equivalence classes of such marked groups $(\Gamma,\Sigma)$ , with $\Sigma=\overline{\Sigma}_{2D}(x)$ , is bounded above if we are able to bound uniformly the cardinality of $\overline{\Sigma}_{2D}(x)$ . But this is an immediate consequence of Proposition 2.3; actually, using the fact that the points in the orbit of $x$ are $\frac{s}{2}$ -separated, we get

\#\overline{\Sigma}_{2D}(x)\leq\text{Pack}\left(2D,\frac{s}{4}\right)\leq P_{0}(1+P_{0})^{\frac{24D}{s}-1}.\qed

Now, the main idea to prove Theorem A is to use the Splitting Theorem 4.1 to show that, up to increasing in a controlled way the codiameter, we can always suppose that the free-systole is bounded away from zero by a universal positive constant: this is the content of the renormalization Theorem E, which is proved below. The proof will show that the space $X^{\prime}$ is isometric to $X$ (though generally the quotients $\Gamma\backslash X$ and $\Gamma\backslash X^{\prime}$ are not isometric, since the first one can be $\varepsilon$ -collapsed for arbitrarily small $\varepsilon$ , while the second one is not collapsed, by construction).
Naively, one can think that if $\textup{sys}^{\diamond}(\Gamma,X)$ is too small then, as we know that $X$ splits $\Gamma$ -invariantly as $Y\times\mathbb{R}^{k}$ by Theorem 4.1, the new space is $X^{\prime}=Y\times s\cdot\mathbb{R}^{k}$ , obtained by dilating the Euclidean factor of a suitable $s>0$ , and the action of $\Gamma$ is the natural one induced on it. The construction of $X^{\prime}$ is however a bit more complicate, since this naïf renormalization is not sufficient in general to enlarge the free systole while keeping the diameter bounded. In order to make things work we need to take into account that $X$ can be collapsed on different subsets at different scales, and an algorithm allowing us to detect them. We refer to Remark 5.2 for a more precise statement.

Recall the constants $n_{0}=P_{0}/2$ , which bounds the dimension of every proper, geodesically complete, CAT $(0)$ -space $X$ which is $(P_{0},r_{0})$ -packed, and $J_{0}=J_{0}(P_{0})$ , introduced at the beginning of Section 4. Also recall the Margulis constant $\varepsilon_{0}=\varepsilon_{0}(P_{0},r_{0})$ given by Proposition 2.4, which we will always assume smaller than $1$ in the sequel.

Proof of Theorem E.

Recall the function $\sigma_{P_{0},r_{0},D_{0}}(\varepsilon)$ of Theorem 4.1. Then we define inductively $D_{1}=2D_{0}+\sqrt{n_{0}}$ , $\sigma_{1}=\sigma_{P_{0},r_{0},D_{1}}(\frac{\varepsilon_{0}}{4J_{0}})$ and

D_{j}=2D_{j-1}+\sqrt{n_{0}}\;\,,\hskip 28.45274pt\sigma_{j}=\sigma_{P_{0},r_{0},D_{j}}(\sigma_{j-1})>0.

We claim that $\Delta_{0}:=D_{n_{0}-1}$ and $s_{0}:=\sigma_{n_{0}}$ satisfy the thesis; notice that both depend only on $P_{0},r_{0}$ and $D_{0}$ .
We now describe a process which takes the CAT $(0)$ -space $X_{0}:=X$ and produces a new proper, geodesically complete, $(P_{0},r_{0})$ -packed, CAT $(0)$ -space $X_{1}$ on which $\Gamma$ still acts faithfully and discretely by isometries, still satisfying all the assumptions of the theorem, except that $\text{diam}(\Gamma\backslash X_{1})\leq D_{1}$ ; and we will show that, repeating again and again this process, we end up with a CAT $(0)$ -space $X_{j}$ with $\textup{sys}^{\diamond}(\Gamma,X_{j})>\sigma_{n_{0}}$ , for some $j\leq n_{0}$ .
If $\textup{sys}^{\diamond}(\Gamma,X_{0})>\sigma_{n_{0}}$ , there is nothing to do, and we just set $X^{\prime}=X_{0}$ .
Otherwise, $\textup{sys}^{\diamond}(\Gamma,X)\leq\sigma_{n_{0}}<\sigma_{1}=\sigma_{P_{0},r_{0},D_{0}}(\frac{\varepsilon_{0}}{4J_{0}})$ , and we apply Theorem 4.1 with $\varepsilon=\frac{\varepsilon_{0}}{4J_{0}}$ . Then, there exists $\varepsilon^{\ast}_{0}:=\varepsilon^{\ast}\in(\sigma_{1},\frac{\varepsilon_{0}}{4J_{0}})$ such that the groups $\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x,X_{0})$ have all rank $k_{0}\geq 1$ for every $x\in X_{0}$ . We then fix $x_{0}\in X_{0}$ . By Proposition 4.3 there exists a free abelian, finite index subgroup $A_{0}$ of $\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x_{0},X)$ of rank $k_{0}\geq 1$ , which is commensurated in $\Gamma$ ; and we have that $X_{0}=\textup{Bd-Min}(\partial A_{0})$ splits isometrically and $\Gamma$ -invariantly as $Y_{0}\times\mathbb{R}^{k_{0}}$ by Proposition 4.5. Moreover, always by Theorem 4.1, there exists $y_{0}\in Y_{0}$ such that $\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ preserves $\{y_{0}\}\times\mathbb{R}^{k_{0}}$ , and there exists a subset of $\overline{\Sigma}_{4J_{0}\cdot\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ whose projection on $\textup{Isom}(\mathbb{R}^{k_{0}})$ generates the maximal lattice $\mathcal{L}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ of the crystallographic group $p_{\mathbb{R}^{k_{0}}}(\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0}))$ . So, we can find a shortest basis $\mathcal{B}_{0}=\{\bf b^{0}_{1},\ldots,\bf b^{0}_{k_{0}}\}$ of the lattice $\mathcal{L}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ whose vectors have all length at most $4J_{0}\cdot\varepsilon^{\ast}_{0}<\varepsilon_{0}<1$ ; without loss of generality, we may suppose that $\|{\bf b^{0}_{1}}\|_{0}\leq\ldots\leq\|{\bf b^{0}_{k_{0}}}\|_{0}=\lambda\left(\mathcal{L}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})\right)=:\ell_{0}<1$ , where $\|\hskip 5.69054pt\|_{0}$ denotes the Euclidean norm of $\mathbb{R}^{k_{0}}$ . By (8), we also know that the covering radius of $\mathcal{L}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ is at most $\frac{\sqrt{k_{0}}}{2}\cdot\ell_{0}\leq\frac{\sqrt{n_{0}}}{2}\cdot\ell_{0}$ .
Now, we define the metric space $X_{1}:=Y_{0}\times\left(\frac{1}{\ell_{0}}\cdot\mathbb{R}^{k_{0}}\right)$ . This is again a proper, geodesically complete, CAT $(0)$ -space, still $(P_{0},r_{0})$ -packed, on which $\Gamma$ still acts discretely by isometries (because the splitting of $X_{0}$ is $\Gamma$ -invariant).We claim that the action of $\Gamma$ on $X_{1}$ is $D_{1}$ -cocompact. In fact, let $x=(y,{\bf v})$ be a point of $X_{1}$ . Since the action of $\Gamma$ on $X_{0}$ is $D_{0}$ -cocompact, we know that there exists $g\in\Gamma$ such that $d_{X_{0}}\left(x,g\cdot(y_{0},{\bf O})\right)\leq D_{0}$ ; moreover, as $\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x_{0},X)$ preserves $\{y_{0}\}\times\mathbb{R}^{k_{0}}$ , we can compose $g$ with elements of this group in order to find $g^{\prime}=(g^{\prime}_{1},g^{\prime}_{2})\in\Gamma$ such that $d_{X_{0}}(x,g^{\prime}\cdot(y_{0},{\bf O}))\leq D_{0}$ and $\|{\bf v}-g^{\prime}_{2}\cdot{\bf O}\|_{0}\leq\frac{\sqrt{n_{0}}}{2}\cdot\ell_{0}$ . Therefore

	$\displaystyle d_{X_{1}}((x,g^{\prime}\cdot(y_{0},{\bf O}))$	$\displaystyle\leq\sqrt{d_{Y_{0}}(y,g_{1}^{\prime}\cdot y_{0})^{2}+\left(\frac{1}{\ell_{0}}\right)^{2}\cdot\\|{\bf v}-g^{\prime}_{2}\cdot{\bf O}\\|_{0}^{2}}$
		$\displaystyle\leq\sqrt{D_{0}^{2}+\frac{n_{0}}{4}}\leq D_{0}+\frac{\sqrt{n_{0}}}{2}=\frac{D_{1}}{2}.$

As $(y,{\bf v})\in X_{1}$ was arbitrary, we then deduce that $X_{1}=\Gamma\cdot\overline{B}_{X_{1}}\left((y_{0},{\bf O}),\frac{D_{1}}{2}\right),$ so $\Gamma<\text{Isom}(X_{1})$ is $D_{1}$ -cocompact. If now $\textup{sys}^{\diamond}(\Gamma,X_{1})>\sigma_{n_{0}}$ , we stop the process and set $X^{\prime}=X_{1}$ : this space has all the desired properties.
Otherwise, we have $\textup{sys}^{\diamond}(\Gamma,X_{1})\leq\sigma_{n_{0}}<\sigma_{2}=\sigma_{P_{0},r_{0},D_{1}}(\sigma_{1})$ and we can apply again Theorem 4.1, Proposition 4.3 and Proposition 4.5 to $X_{1}$ , with $\varepsilon=\sigma_{1}$ . Then, there exists $\varepsilon^{\ast}_{1}\in(\sigma_{2},\sigma_{1})$ such that the groups $\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x,X_{1})$ have rank $k_{1}\geq 1$ for every $x\in X_{1}$ , in particular $\text{rk}(\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1}))=k_{1}$ . Moreover, there exists a free abelian, finite index subgroup $A_{1}<\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})$ of rank $k_{1}$ which is commensurated in $\Gamma$ , the space $X_{1}=\textup{Bd-Min}(\partial A_{1})$ splits isometrically and $\Gamma$ -invariantly as $Y_{1}\times\mathbb{R}^{k_{1}}$ , and the trace at infinity $\partial\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})$ coincides with the boundary at infinity of all the sets $\{y\}\times\mathbb{R}^{k_{1}}$ ; and there exists a subset of $\overline{\Sigma}_{4J_{0}\cdot\varepsilon^{\ast}_{1}}(x_{0},X_{1})$ whose projection on $\textup{Isom}(\mathbb{R}^{k_{1}})$ generates the maximal lattice $\mathcal{L}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})$ of $p_{\mathbb{R}^{k_{1}}}(\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1}))$ . Therefore, we can find a shortest basis $\mathcal{B}^{1}=\{\bf b_{1}^{1},\ldots,\bf b_{k_{1}}^{1}\}$ of $\mathcal{L}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})$ with lengths (with respect to the Euclidean norm $\|\hskip 5.69054pt\|_{1}$ of $\mathbb{R}^{k_{1}}$ )

\|{\bf b_{1}^{1}}\|_{1}\leq\ldots\leq\|{\bf b_{k_{1}}^{1}}\|_{1}=\lambda\left(\mathcal{L}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})\right)=:\ell_{1}\leq 4J_{0}\cdot\sigma_{1}<\varepsilon_{0}<1.

Observe that, as by construction the factor $\left(\frac{1}{\ell_{0}}\right)^{2}$ is bigger than $1$ , we have

(9)

\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})<\overline{\Gamma}_{\sigma_{1}}(x_{0},X_{1})<\overline{\Gamma}_{\sigma_{1}}(x_{0},X_{0})<\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0}),

so $k_{1}=\text{rk}(\overline{\Gamma}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1}))\leq k_{0}=\text{rk}(\overline{\Gamma}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0}))$ , and $A_{1}<A_{0}$ . In particular, the metric splittings of $X_{1}$ as $Y_{0}\times\left(\frac{1}{\ell_{0}}{\mathbb{R}}^{k_{0}}\right)$ and $Y_{1}\times{\mathbb{R}}^{k_{1}}$ determined, respectively, by the groups $A_{0}$ and $A_{1}$ satisfy ${\mathbb{R}}^{k_{1}}\subset\frac{1}{\ell_{0}}{\mathbb{R}}^{k_{0}}$ and $Y_{0}\subset Y_{1}$ , because of the second part of Proposition 4.5.
We will now show that $k_{1}\lneq k_{0}$ . Actually, suppose that $k_{1}=k_{0}=:k$ . Then, the groups $A_{0}$ and $A_{1}$ split the same Euclidean factor and $Y_{0}=Y_{1}$ . The lengths of the basis $\mathcal{B}^{1}$ with respect to the Euclidean norm $\|\hskip 5.69054pt\|_{0}$ of the Euclidean factor ${\mathbb{R}}^{k_{0}}$ of $X_{0}$ are $\|{\bf b_{i}^{1}}\|_{0}=\ell_{0}\cdot\|{\bf b_{i}^{1}}\|_{1}<\ell_{0}$ for every $i=1,\ldots,k$ . But then, since $\mathcal{L}_{\varepsilon^{\ast}_{1}}(x_{0},X_{1})<\mathcal{L}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ by (9), we would be able to find $k$ independent vectors of $\mathcal{L}_{\varepsilon^{\ast}_{0}}(x_{0},X_{0})$ of length less than its shortest generating radius, which is impossible. This shows that $k_{1}<k_{0}$ .
We can now define $X_{2}:=Y_{1}\!\times\left(\frac{1}{\ell_{1}}\!\cdot\mathbb{R}^{k_{1}}\right)$ , on which $\Gamma$ acts faithfully, discretely by isometries, $D_{2}$ -cocompactly, for $D_{2}=2D_{1}+\sqrt{n_{0}}$ computed as before. We can repeat this process to get a sequence of proper, geodesically complete, $(P_{0},r_{0})$ -packed, CAT $(0)$ -spaces $X_{j}$ on which $\Gamma$ always acts faithfully, discretely and $D_{j}$ -cocompactly by isometries. Moreover at each step either $\textup{sys}^{\diamond}(\Gamma,X_{j})\geq\sigma_{n_{0}}$ or the $\varepsilon^{\ast}_{j}$ -splitting rank $k_{j}$ of $X_{j}$ provided by Theorem 4.1 is strictly smaller than the $\varepsilon^{\ast}_{j-1}$ -splitting rank $k_{j-1}$ of $X_{j-1}$ . Since the splitting rank of $X_{0}$ is at most $n_{0}$ there must exist $j\in\{1,\ldots,n_{0}\}$ such that $\textup{sys}^{\diamond}(\Gamma,X_{j})\geq\sigma_{n_{0}}$ . The proof then ends by setting $X^{\prime}=X_{j}$ .
Clearly, we may take, explicitely, $\Delta_{0}=2^{n_{0}}D_{0}+(2^{n_{0}}-1)\sqrt{n_{0}}$ .
Finally, observe that, by construction, $X^{\prime}$ is isometric to the initial space $X$ , and that the action of $\Gamma$ on $X_{j}$ is nonsingular if and only if the action of $\Gamma$ on $X_{j-1}$ , and by induction on $X$ , was nonsingular. ∎

Remark 5.2.

The proof of Theorem E actually gives us something more. Indeed we produce a sequence of free abelian commensurated subgroups of $\Gamma$

\{\textup{id}\}\lneqq A_{0}\lneqq A_{1}\ldots\lneqq A_{m}

for some $m\leq n_{0}-1$ , such that:

(a)

denoting by $k_{j}$ the rank of $A_{j}$ , then $1\leq k_{0}<k_{1}\ldots<k_{m}$ ;

(b)

setting $h_{j}=k_{j}-k_{j-1}$ then there is a corresponding isometric and $\Gamma$ -invariant splitting of $X$ as $Y\times\mathbb{R}^{h_{0}}\times\mathbb{R}^{h_{1}}\times\cdots\times\mathbb{R}^{h_{m}}$ . Moreover, there exist $0<L_{j}<1$ such that the natural action of $\Gamma$ on the space

(10)

X^{\prime}:=Y\times\left(\frac{1}{L_{0}}\cdot\mathbb{R}^{h_{0}}\right)\times\left(\frac{1}{L_{1}}\cdot\mathbb{R}^{h_{1}}\right)\times\cdots\times\left(\frac{1}{L_{m}}\cdot\mathbb{R}^{h_{m}}\right)

is $\Delta_{0}$ -cocompact with free-systole at least $s_{0}$ .

The finiteness Theorem A is now an immediate consequence of the above renormalization result, combined with Proposition 5.1 and Theorem 3.1:

Proof of Theorem A.

By Theorem E every group $\Gamma$ under consideration acts discretely by isometries and $\Delta_{0}$ -cocompactly on a proper, geodesically complete, $(P_{0},r_{0})$ -packed, CAT $(0)$ -space $X^{\prime}$ with $\textup{sys}^{\diamond}(\Gamma,X^{\prime})\geq s_{0}$ . The action is still nonsingular, so by Theorem 3.1 we can find $x^{\prime}\in X^{\prime}$ with $\textup{sys}(\Gamma,x^{\prime})\geq b_{0}\cdot s_{0}$ , a positive lower bound depending only on $P_{0},r_{0}$ and $D_{0}$ . Then, applying Proposition 5.1 with $D_{0}=\Delta_{0}=D$ and $s=b_{0}\cdot s_{0}$ , we conclude the proof.

∎

Remark 5.3.

The assumption that the groups $\Gamma$ act nonsingularly is necessary for Theorem A. Actually, in [BK90, Theorem 7.1], Bass and Kulkarni exhibit an infinite ascending family of discrete groups $\Gamma_{1}<\Gamma_{2}<\cdots$ acting (singularly) on a regular tree $X$ with bounded valency, with same compact quotient, in particular with diam $(\Gamma_{j}\backslash X)\leq 2$ for all $j$ . Moreover these groups are lattices satisfying Vol $(\Gamma_{i}\backslash\backslash X)\rightarrow 0$ , where the volume of $\Gamma$ in $X$ is defined as

\text{Vol}(\Gamma\backslash\backslash X)=\sum_{x\in\Gamma\backslash X}\frac{1}{|\text{Stab}_{\Gamma}(x)|}.

This family contains infinitely many different groups, since the minimal order $\sigma_{i}$ of a torsion subgroup in $\Gamma_{i}$ tends to infinity as the volume goes to zero (every torsion subgroup of $\Gamma_{i}$ stabilizes some point, as $X$ is CAT $(0)$ , hence $\frac{1}{\sigma_{i}}$ is the dominant term of the sum yielding $\text{Vol}(\Gamma_{i}\backslash\backslash X)$ ).

Proof of Corollary F.

The number of isomorphism types of possible group is finite by Theorem B, so the thesis is a trivial consequence of this theorem. However we can give a more costructive proof that gives an explicit upper bound for the order of finite subgroups. By Theorem E we can suppose that $\Gamma$ is $\Delta_{0}$ -cocompact and $\text{sys}^{\diamond}(\Gamma,X)\geq s_{0}$ . Let $F<\Gamma$ be a finite subgroup. It fixes a point $x_{0}\in X$ (cp. [BH13, Corollary II.2.8]). Let $\mathrm{D}_{x_{0}}$ be the Dirichlet domain of $\Gamma$ at $x_{0}$ , which is clearly $F$ -invariant. By Theorem 3.1 there exists some point $x\in\mathrm{D}_{x_{0}}$ where $\text{sys}(\Gamma,x)\geq b_{0}\cdot s_{0}>0$ . Therefore the orbit $\{fx\}_{f\in F}$ is a $\frac{b_{0}\cdot s_{0}}{2}$ -separated subset of $\mathrm{D}_{x_{0}}$ and the balls $B(fx,\frac{b_{0}\cdot s_{0}}{4})$ are all disjoint. Using the fact that $\mathrm{D}_{x_{0}}$ is contained in $\overline{B}(x_{0},\Delta_{0})$ we get

V(\Delta_{0})\geq\mu_{X}(\mathrm{D}_{x_{0}})\geq\#F\cdot v\left(\frac{b_{0}\cdot s_{0}}{4}\right),

by Proposition 2.3.(iii). This yields the explicit bound $\#F\leq\frac{V(\Delta_{0})}{v\left(\frac{b_{0}\cdot s_{0}}{4}\right)}$ for the cardinality of $F$ , only depending only on $P_{0},r_{0}$ and $D_{0}$ . ∎

5.2. Finiteness up to equivariant homotopy equivalence

We have proved so far that the number of discrete, nonsingular and $D_{0}$ -cocompact groups of isometries $\Gamma$ of proper, geodesically complete, $(P_{0},r_{0})$ -packed, CAT $(0)$ -spaces $X$ is finite up to isomorphism of abstract groups.
In this section we will show the finiteness up to equivariant homotopy equivalence of orbispaces. That is, we can upgrade every isomorphism of groups $\varphi:\Gamma\rightarrow\Gamma^{\prime}$ , acting discretely, nonsingularly and cocompactly by isometries respectively on CAT $(0)$ -spaces $X$ and $X^{\prime}$ , to a homotopy equivalence $F:X\rightarrow X^{\prime}$ which is $\varphi$ -equivariant, i.e. $F(g\cdot x)=\varphi(g)\cdot F(x)$ for all $g\in\Gamma$ and all $x\in X$ . Namely we show:

Proposition 5.4.

Let $\Gamma$ and $\Gamma^{\prime}$ be discrete, cocompact, nonsingular isometry groups of two proper, geodesically complete, $(P_{0},r_{0})$ -packed, $\textup{CAT}(0)$ -spaces $X$ and $X^{\prime}$ , respectively. If there exists a group isomorphism $\varphi:\Gamma\rightarrow\Gamma^{\prime}$ , then there exists a $\varphi$ -equivariant homotopy equivalence $f:X\rightarrow X^{\prime}$ .

This is a CAT(0) version of a result about the realization of orbifold group-isomorphisms by orbi-maps equivalences, see [Yam90, Theorem 2.5]. We will give a direct proof of this fact, using the barycenter technique introduced by Besson-Courtois-Gallot [BCG95] for strictly negatively curved manifolds (cp. [Sam99] for an approach similar to the one we follow here).

Let $X$ be a proper, CAT $(0)$ -metric space. Consider the space ${\mathcal{M}}_{2}(X)$ of positive, finite, Borel measures $\mu$ on $X$ with finite second moment, i.e. such that the distance function $d(x,\cdot)$ is $\mu$ -square integrable for some (hence every) $x\in X$ . The barycenter of $\mu\in{\mathcal{M}_{2}}(X)$ is defined as the unique point of minimum of the function

\mathcal{B}_{\mu}(x)=\int_{X}d(x,y)^{2}\,d\mu(y).

Notice that $\mathcal{B}_{\mu}(x)$ tends to $+\infty$ for $x\rightarrow\infty$ since for every fixed $x_{0}$ in $X$ we have, by triangular and Schwarz inequalities,

\mathcal{B}_{\mu}(x)\geq d(x,x_{0})^{2}\mu(X)-2d(x,x_{0})\mu(X)\mathcal{B}_{\mu}(x_{0})+\mathcal{B}_{\mu}(x_{0})

which diverges as $d(x,x_{0})\rightarrow+\infty$ . Moreover, by standard comparison with the Euclidean space, the function $d(x,\cdot)^{2}$ is also strictly convex, namely $2$ -convex in the sense of [Kle99]: that is, $d(x,c(t))-t^{2}$ is a convex function, for every geodesic $c:[a,b]\rightarrow X$ . It follows that the function $\mathcal{B}_{\mu}$ is $2\mu(X)$ -convex as well, which implies that $\mathcal{B}_{\mu}$ has a unique point of minimum (cp. [Kle99, Lemma 2.3]). Therefore $\texttt{bar}[\mu]:=\text{arg}\,\text{min}(\mathcal{B}_{\mu})$ is well-defined.
It is straightforward to check that the barycenter is equivariant with respect to the natural actions of $\text{Isom}(X)$ on $X$ and ${\mathcal{M}}_{2}(X)$ , that is:

(11)

\texttt{bar}[g_{\ast}\mu]=g\cdot\texttt{bar}[\mu],\hskip 5.69054pt\forall g\in\text{Isom}(X),\forall\mu\in{\mathcal{M}}_{2}(X)

We can now give the Proof of Proposition 5.4:

Proof of Proposition 5.4..

As $(\Gamma,X)$ and $(\Gamma^{\prime},X^{\prime})$ are nonsingular, we can choose $x_{0}\in X$ and $x_{0}^{\prime}\in X^{\prime}$ such that the stabilizers $\text{Stab}_{\Gamma}(x_{0})$ , $\text{Stab}_{\Gamma^{\prime}}(x_{0}^{\prime})$ are trivial. Then there exists a unique $\varphi$ -equivariant map $\phi:\Gamma x_{0}\rightarrow\Gamma^{\prime}x_{0}^{\prime}$ sending $x_{0}$ to $x_{0}^{\prime}$ , namely $\phi(g\cdot x_{0})=\varphi(g)\cdot x_{0}^{\prime}$ . Now, by $\check{\textup{S}}$ varc-Milnor Lemma, the spaces $X$ and $X^{\prime}$ are respectively quasi-isometric to the orbits $\Gamma x_{0}$ and $\Gamma^{\prime}x_{0}^{\prime}$ and also to the marked groups $(\Gamma,\Sigma_{2D}(x_{0}))$ and $(\Gamma^{\prime},\Sigma_{2D}(x_{0}^{\prime}))$ endowed with their word metrics. Moreover, since the groups $\Gamma$ and $\Gamma^{\prime}$ are isomorphic, and any two word metrics associated to finite generating sets on the same group are equivalent, we deduce that the map $\phi$ is an $(a,b)$ -quasi-isometry for suitable $a,b$ , in particular

(12)

d(\varphi(g_{1})x_{0}^{\prime},\varphi(g_{2})x_{0}^{\prime})\leq ad(g_{1}x_{0},g_{2}x_{0})+b\hskip 14.22636pt\text{ for all }g_{1},g_{2}\in\Gamma

We now define a $\varphi$ -equivariant homotopy equivalence $f:X\rightarrow X^{\prime}$ as follows: choose any $h>\frac{\log(1+P_{0})}{r_{0}}$ (the upper bound of the entropy of $X$ given by Proposition 2.3.(iv) and consider the family of measures $\mu_{x}\in{\mathcal{M}}_{2}(X^{\prime})$ , indexed by $x\in X$ and supported by the orbit of $x_{0}^{\prime}$ , given by

\mu_{x}:=\sum_{g\in\Gamma}e^{-hd(x,gx_{0})}\delta_{\varphi(g)x_{0}^{\prime}},

where $\delta_{\varphi(g)x_{0}^{\prime}}$ is the Dirac measure at $\varphi(g)x_{0}^{\prime}$ , and then define

f(x):=\texttt{bar}[\mu_{x}].

Notice that the total mass of $\mu_{x}$ coincides with the value of the Poincaré series $P_{\Gamma}(x,x_{0},s)=\sum_{g\in\Gamma}e^{-sd(x,gx_{0})}$ of the group $\Gamma$ acting on $X$ for $s=h$ , which is finite since $h$ is chosen greater than the critical exponent of the series (recall that $\text{Ent}(X)$ equals the critical exponent of the group $\Gamma$ acting on $X$ , since the action is cocompact, see for instance [BCGS17] or [Cav22a, Proposition 5.7]). Moreover, the function $d(x_{0}^{\prime},\cdot)$ is square summable with respect to $\mu_{x}$ , since we have by (12)

	$\displaystyle\int_{X^{\prime}}d(x_{0}^{\prime},y)^{2}d\mu_{x}(y)$	$\displaystyle=\sum_{g\in\Gamma}d(x_{0}^{\prime},\varphi(g)x_{0}^{\prime})^{2}e^{-hd(x,gx_{0})}$
		$\displaystyle\leq\sum_{g\in\Gamma}(a\cdot d(x_{0},gx_{0})+b)^{2}e^{-hd(x,gx_{0})}$

which is finite as the Poincaré series converges exponentially fast for $s=h$ .
Therefore the map $f$ is well-defined.

Step 1: $f$ is continuous.
First, we show that $\mathcal{B}_{\mu_{x_{n}}}$ converge uniformly on compacts to $\mathcal{B}_{\mu_{x}}$ for $x_{n}\rightarrow x$ in $X$ . Actually, let $K\subset X^{\prime}$ be a fixed compact subset. Since the action of $\Gamma^{\prime}$ on $X^{\prime}$ is discrete, the subset

S=\{g^{\prime}\in\Gamma^{\prime}\hskip 2.84526pt|\hskip 2.84526ptg^{\prime}B(x_{0}^{\prime},2D_{0})\cap K\neq\emptyset\}

is finite. Then for every $x^{\prime}\in K$ choose some $g_{x^{\prime}}\in\Gamma$ with $\varphi(g_{x^{\prime}})\in S$ such that $d(x^{\prime},\varphi(g_{x^{\prime}})x_{0}^{\prime})\leq 2D_{0}$ . From (12) and the triangular inequality we find that, for $d(x_{n},x)<\varepsilon$ , it holds

\left|\mathcal{B}_{\mu_{x_{n}}}(x^{\prime})-\mathcal{B}_{\mu_{x}}(x^{\prime})\right|\leq\sum_{g\in\Gamma}d(x^{\prime},\varphi(g)\cdot x_{0}^{\prime})^{2}\left|e^{-hd(x_{n},g\cdot x_{0})}-e^{-hd(x,g\cdot x_{0})}\right|\hskip 28.45274pt

\hskip 42.67912pt\leq h\left(\sum_{g\in\Gamma}\left(d(\varphi(g_{x^{\prime}})x_{0}^{\prime},\varphi(g)x_{0}^{\prime})+2D_{0}\right)^{2}e^{-h\left(d(x,gx_{0})-\varepsilon\right)}\right)d(x_{n},x)

\leq h\left(\sum_{g\in\Gamma}\left(a\cdot d(x_{0},g\cdot x_{0})+b^{\prime}\right)^{2}e^{-h\left(d(x,g_{x^{\prime}}gx_{0})-\varepsilon\right)}\right)d(x_{n},x)

\hskip 36.98857pt\leq he^{h\varepsilon}e^{hd(x,g_{x^{\prime}}x)}\left(\sum_{g\in\Gamma}\left(a\cdot d(x_{0},g\cdot x_{0})+b^{\prime}\right)^{2}e^{-hd(x,g\cdot x_{0})}\right)d(x_{n},x)

for $b^{\prime}=b+2D_{0}$ . Now, the series in parentheses is bounded above independently of $n$ and $x^{\prime}$ , while (for fixed $x$ ) the term $e^{hd(x,g_{x^{\prime}}\cdot x)}$ is uniformly bounded on $K$ since $g_{x^{\prime}}$ belong to the finite subset $\varphi^{-1}(S)$ . The above estimate then implies that $\mathcal{B}_{\mu_{x_{n}}}\rightarrow\mathcal{B}_{\mu_{x}}$ uniformily on $K$ when $x_{n}\rightarrow x$ .
Secondly, we call $m=\min\mathcal{B}_{\mu_{x}}$ and show that there exists $R$ such that for every $n\gg 0$ the functions $\mathcal{B}_{\mu_{x_{n}}}$ are greater than $2m$ on $X^{\prime}\setminus\overline{B}(\texttt{bar}[\mu_{x}],R)$ .
Actually, let $R$ be such that $\mathcal{B}_{\mu_{x}}>2m$ outside $B(\texttt{bar}[\mu_{x}],R/2)$ (recall that $\mathcal{B}_{\mu_{x}}$ is proper, since we showed that $\lim_{x^{\prime}\rightarrow\infty}\mathcal{B}_{\mu}(x^{\prime})=+\infty$ for all $\mu\in{\mathcal{M}}_{2}(X)$ ), and assume that we have infinitely many points $x_{n}^{\prime}$ with $d(\texttt{bar}[\mu_{x}],x_{n}^{\prime})>R$ such that $\mathcal{B}_{\mu_{x_{n}}}(x_{n}^{\prime})\leq 2m$ . Then, calling $y^{\prime}_{n}\in[\texttt{bar}[\mu_{x}],x^{\prime}_{n}]$ the point at distance $R$ from $\texttt{bar}[\mu_{x}]$ , we would have, by convexity,

\mathcal{B}_{\mu_{x_{n}}}(y^{\prime}_{n})\leq\max\{\mathcal{B}_{\mu_{x_{n}}}(\texttt{bar}[\mu_{x}]),2m\}.

Since $\mathcal{B}_{\mu_{x_{n}}}\rightarrow\mathcal{B}_{\mu_{x}}$ uniformly on $\overline{B}(\texttt{bar}[\mu_{x}],R)$ we would deduce that $\mathcal{B}_{\mu_{x}}(y_{n}^{\prime})\leq 2m$ , which contradicts the choice of $R$ , since $d(\texttt{bar}[\mu_{x}],y_{n}^{\prime})>R/2$ .
Now, since $\mathcal{B}_{\mu_{x_{n}}}>2m$ on $X^{\prime}\setminus\overline{B}(\texttt{bar}[\mu_{x}],R)$ for all $n\gg 0$ , the uniform convergence $\mathcal{B}_{\mu_{x_{n}}}\rightarrow\mathcal{B}_{\mu_{x}}$ on $\overline{B}(\texttt{bar}[\mu_{x}],R)$ implies that the sequence $f(x_{n})=\texttt{bar}[\mu_{x_{n}}]$ of (unique) minimum points of $\mathcal{B}_{\mu_{x_{n}}}$ converge to the (unique) minimum point $f(x)=\texttt{bar}[\mu_{x}]$ of $\mathcal{B}_{\mu_{x}}$ . In fact, we have that $\texttt{bar}[\mu_{x_{n}}]\in\overline{B}(\texttt{bar}[\mu_{x}],R)$ for all $n\gg 0$ , and

(13)

\mathcal{B}_{\mu_{x_{n}}}(\texttt{bar}[\mu_{x}])\geq\mathcal{B}_{\mu_{x_{n}}}(\texttt{bar}[\mu_{x_{n}}])

so if $\texttt{bar}[\mu_{x_{n}}]$ accumulates to a point $b_{\infty}$ , passing to the limit in (13) yields $\mathcal{B}_{\mu_{x}}(\texttt{bar}[\mu_{x}])\geq\mathcal{B}_{\mu_{x}}(b_{\infty})$ . This implies that $b_{\infty}=\texttt{bar}[\mu_{x}]$ , by unicity of the minimum point of $\mathcal{B}_{\mu_{x}}$ .

Step 2: $f$ is a $\varphi$ -equivariant homotopy equivalence.
Firstly, the map $f$ is $\varphi$ -equivariant, since for all $g\in\Gamma$ we have

\varphi(g)_{\!\ast}\,\mu_{x}=\sum_{\gamma\in\Gamma}e^{-hd(x,\gamma x_{0})}\delta_{\varphi(g\gamma)\cdot x_{0}^{\prime}}=\sum_{\gamma\in\Gamma}e^{-hd(x,g^{-1}\gamma x_{0})}\delta_{\varphi(\gamma)x_{0}^{\prime}}=\mu_{gx}

and by (11) we deduce $f(g\cdot x)=\texttt{bar}[\varphi(g)_{\ast}\mu_{x}]=\varphi(g)\cdot\texttt{bar}[\mu_{x}]=\varphi(g)\cdot f(x)$ .
Then, from the inverse map $\phi^{-1}:\Gamma^{\prime}x_{0}^{\prime}\rightarrow\Gamma x_{0}$ we can analogously construct a $\varphi^{-1}$ -equivariant, continuous map $f^{\prime}:X^{\prime}\rightarrow X$ . Now consider the homotopy map $H:X\times[0,1]\rightarrow X$ defined by the formula $H(x,t)=tx+(1-t)\,f^{\prime}\!\circ\!f(x)$ (where $tx+(1-t)y$ denotes the point on the geodesic segment $[x,y]$ at distance $t/d(x,y)$ from $x$ ). The composition $f^{\prime}\circ f$ is $\Gamma$ -equivariant, so we deduce that $H$ is a $\Gamma$ -equivariant homotopy between $f^{\prime}\circ f$ and $\text{id}_{X}$ .
Similarly, one proves that the map $H^{\prime}(x^{\prime},t)=tx^{\prime}+(1-t)\,f\!\circ\!f^{\prime}(x)$ is a $\Gamma^{\prime}$ -equivariant homotopy between $f\circ f^{\prime}$ and $\text{id}_{X^{\prime}}$ . ∎

5.3. Finiteness of nonpositively curved orbifolds

We restrict here our attention to CAT $(0)$ -orbispaces which are quotients, by a discrete isometry group $\Gamma$ , of a Hadamard manifold $X$ (that is, a complete, connected and simply connected, nonpositively curved Riemannian manifold) with pinched sectional curvature $-\kappa^{2}\leq k(X)\leq 0$ : we call such a quotient $M=\Gamma\backslash X$ a nonpositively curved Riemannian orbifold with curvature $k(M)\geq-\kappa^{2}$ . Recall that, in this case, any discrete isometry group $\Gamma$ of $X$ is rigid, in the sense explained in Section 2.1.

For compact, non-positively curved Riemannian orbifolds, the finiteness up to equivariant homotopy equivalence can be improved to finiteness up to equivariant diffeomorphisms: recall that an equivariant diffeomorphism between two Riemannian orbifolds $M=\Gamma\backslash X$ , $M^{\prime}=\Gamma^{\prime}\backslash X^{\prime}$ is a diffeomorphism $F:X\rightarrow X^{\prime}$ which is equivariant with respect to some group isomorphism $\varphi:\Gamma\rightarrow\Gamma^{\prime}$ , i.e. $F(g\cdot x)=\varphi(g)\cdot F(x)$ for all $g\in\Gamma$ and all $x\in X$ .

Proof of Corollary C.

First notice that the splitting Theorem D is true in the manifold category. Actually, let $X={\mathbb{R}}^{n}\times X_{0}$ be the De Rham decomposition of a Hadamard manifold $X$ , where $X_{0}$ is the product of all non-Euclidean factors. The factor ${\mathbb{R}}^{n}$ coincides with the Euclidean factor splitted by $X$ as a CAT $(0)$ -space (since the decomposition of a finite dimensional geodesic space into flat and irreducible factors is invariant by isometries, cp. [FL08]). Then, the ${\mathbb{R}}^{k}$ factor splitted by Theorem D under an $\varepsilon$ -collapsed action, with $\varepsilon\leq\sigma_{0}$ , is isometrically immersed in the Euclidean de Rham factor ${\mathbb{R}}^{n}$ of the manifold decomposition of $X$ ; hence it is $C^{\infty}$ -embedded as a submanifold, as well as its orthogonal complement ${\mathbb{R}}^{m-k}$ .Then, also the factor $Y$ of Theorem D is $C^{\infty}$ -embedded in $X$ , because $Y=X_{0}\times{\mathbb{R}}^{m-k}$ . From this, we deduce that also the renormalization Theorem E holds in the manifold category: that is, every splitting $Y_{i}\times{\mathbb{R}}^{k_{i}}$ considered in the proof is smooth, and the resulting decomposition (10) yields a smooth Riemannian structure on $X^{\prime}$ (such that $X^{\prime}=X$ as a differentiable manifold). Moreover, if $-\kappa^{2}\leq k(X)\leq 0$ , the new Riemannian manifold $X^{\prime}$ satisfies the same curvature bounds (since the metric is dilated on the Euclidean factors by the constants $\frac{1}{L_{i}}>1$ , as explained in Remark 5.2).
It then follows that every $n$ -dimensional Riemannian orbifold $M=\Gamma\backslash X$ with $-\kappa^{2}\leq k(M)\leq 0$ and diam $(M)\leq D_{0}$ is equivariantly diffeomorphic to a Riemannian orbifold $M^{\prime}=\Gamma\backslash X^{\prime}$ still satisfying $-\kappa^{2}\leq k(M^{\prime})\leq 0$ , but with diameter bounded above by $\Delta_{0}$ and with sys ${}^{\diamond}(\Gamma,X^{\prime})\geq s_{0}$ ; where the constants $\Delta_{0}$ and $s_{0}$ only depend on $D_{0}$ and on the packing constants $(P_{0},r_{0})$ of $X$ (and then, ultimately, only on $\kappa$ and on the dimension $n$ , by (6)).Moreover, by Theorem 3.1, the systole of $\Gamma$ acting on $X^{\prime}$ is bounded below by $b_{0}\cdot\min\{\text{sys}^{\diamond}(\Gamma,X^{\prime}),\varepsilon_{0}\}$ at some point $x^{\prime}$ , where again $b_{0},\varepsilon_{0}$ only depend on $(P_{0},r_{0})$ . We then deduce the finiteness of these orbifolds from Fukaya’s [Fuk86, Theorem 8.1] (or from Cheeger’s finiteness theorem, as completed in [Pet84], [Yam85], in the torsion-free case). ∎

References

[BCG95] G. Besson, G. Courtois, and S. Gallot. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geometric and Functional Analysis, 5:731–799, 09 1995.
[BCGS17] G. Besson, G. Courtois, S. Gallot, and A. Sambusetti. Curvature-free Margulis lemma for Gromov-hyperbolic spaces. arXiv preprint arXiv:1712.08386, 2017.
[BCGS21] G. Besson, G. Courtois, S. Gallot, and A. Sambusetti. Finiteness theorems for Gromov-hyperbolic spaces and groups. arXiv preprint arXiv:2109.13025, 2021.
[BGT11] E. Breuillard, B. Green, and T. Tao. The structure of approximate groups. Publications mathématiques de l’IHÉS, 116, 10 2011.
[BH13] M. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319. Springer Science & Business Media, 2013.
[BK90] H. Bass and R. Kulkarni. Uniform tree lattices. Journal of the American Mathematical Society, 3(4):843–902, 1990.
[Buy81] S. Buyalo. Manifolds of nonpositive curvature with small volume. Mathematical notes of the Academy of Sciences of the USSR, 29(2):125–130, 1981.
[Buy83] S. Buyalo. Volume and fundamental group of a manifold of nonpositive curvature. Mat. Sb. (N.S.), 122(164)(2):142–156, 1983.
[Buy90a] S. Buyalo. Collapsing manifolds of non-positive curvature I. Leningrad Math. J., 1, No. 5:1135?1155, 1990.
[Buy90b] S. Buyalo. Collapsing manifolds of non-positive curvature II. Leningrad Math. J., 1, No. 6:1371?1399, 1990.
[Cav21] N. Cavallucci. Topological entropy of the geodesic flow of non-positively curved metric spaces. arXiv preprint arXiv:2105.11774, 2021.
[Cav22a] N. Cavallucci. Continuity of critical exponent of quasiconvex-cocompact groups under Gromov–Hausdorff convergence. Ergodic Theory and Dynamical Systems, pages 1–33, 2022.
[Cav22b] N. Cavallucci. Entropies of non-positively curved metric spaces. Geometriae Dedicata, 216(5):54, 2022.
[CCR01] J. Cao, J. Cheeger, and X. Rong. Splittings and cr-structures for manifolds with nonpositive sectional curvature. Inventiones mathematicae, 144(1):139–167, 2001.
[CFG92] J. Cheeger, K. Fukaya, and M. Gromov. Nilpotent structures and invariant metrics on collapsed manifolds. J. Amer. Math. Soc., 5(2):327–372, 1992.
[CG86] J. Cheeger and M. Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded I. J. Diff. Geom., 23:309–364, 1986.
[CG90] J. Cheeger and M. Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded. II. J. Differential Geom., 32(1):269–298, 1990.
[Che70] J. Cheeger. Finiteness theorems for Riemannian manifolds. Amer. J. Math., 92:61–74, 1970.
[CM09a] P.E. Caprace and N. Monod. Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol., 2(4):701–746, 2009.
[CM09b] P.E. Caprace and N. Monod. Isometry groups of non-positively curved spaces: structure theory. Journal of topology, 2(4):661–700, 2009.
[CM19] P.E. Caprace and N. Monod. Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups". arXiv preprint arXiv:1908.10216, 2019.
[CS20] N. Cavallucci and A. Sambusetti. Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces. arXiv preprint arXiv:2102.09829, 2020.
[CS21] N. Cavallucci and A. Sambusetti. Packing and doubling in metric spaces with curvature bounded above. Mathematische Zeitschrift, pages 1–46, 2021.
[CS22] N. Cavallucci and A. Sambusetti. Thin actions on CAT(0) spaces. arXiv preprint arXiv:2210.01085, 2022.
[CS23] N. Cavallucci and A. Sambusetti. Convergence and collapsing of CAT $(0)$ -group actions. arXiv preprint arXiv:2304.10763, 2023.
[DLHG90] P. De La Harpe and E. Ghys. Espaces métriques hyperboliques. Sur les groupes hyperboliques d’après Mikhael Gromov, pages 27–45, 1990.
[DY05] F. Dahmani and A. Yaman. Bounded geometry in relatively hyperbolic groups. New York J. Math., 11:89–95, 2005.
[Ebe88] P. Eberlein. Symmetry diffeomorphism group of a manifold of nonpositive curvature. Trans. Amer. Math. Soc., 309(1):355–374, 1988.
[Far81] D.R. Farkas. Crystallographic groups and their mathematics. Rocky Mountain J. Math., 11(4):511–551, 1981.
[FL08] T. Foertsch and A. Lytchak. The de Rham decomposition theorem for metric spaces. Geom. Funct. Anal., 18:120–143, 2008.
[Fuk86] K. Fukaya. Theory of convergence for Riemannian orbifolds. Japanese journal of mathematics. New series, 12(1):121–160, 1986.
[Fuk87] K. Fukaya. Collapsing Riemannian manifolds to ones of lower dimensions. J. Differential Geom., 25(1):139–156, 1987.
[Fuk88] K. Fukaya. A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. J. Differential Geom., 28(1):1–21, 1988.
[Gro78] M. Gromov. Manifolds of negative curvature. Journal of differential geometry, 13(2):223–230, 1978.
[Gro81] M. Gromov. Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits). Publications Mathématiques de l’IHÉS, 53:53–78, 1981.
[Gro07] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Basel: Birkhäuser, 2007.
[Gut10] L. Guth. Metaphors in systolic geometry. In Proceedings of the International Congress of Mathematicians. Volume II, pages 745–768. Hindustan Book Agency, New Delhi, 2010.
[Kle99] B. Kleiner. The local structure of length spaces with curvature bounded above. Mathematische Zeitschrift, 231, 01 1999.
[LN19] A. Lytchak and K. Nagano. Geodesically complete spaces with an upper curvature bound. Geometric and Functional Analysis, 29(1):295–342, Feb 2019.
[Pet84] S. Peters. Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds. J. Reine Angew. Math., 349:77–82, 1984.
[PT99] A. Petrunin and W. Tuschmann. Diffeomorphism finiteness, positive pinching, and second homotopy. Geom. Funct. Anal., 9(4):736–774, 1999.
[Sab20] S. Sabourau. Macroscopic scalar curvature and local collapsing. arXiv preprint arXiv:2006.00663, 2020.
[Sam99] A. Sambusetti. Minimal entropy and simplicial volume. Manuscripta Mathematica, 99(4):541–560, 1999.
[Ser03] J.P. Serre. Trees. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.
[Stu06] Karl-Theodor Sturm. On the geometry of metric measure spaces. II. Acta Math., 196(1):133–177, 2006.
[Wei67] A. Weinstein. On the homotopy type of positively-pinched manifolds. Arch. Math. (Basel), 18:523–524, 1967.
[Yam85] T. Yamaguchi. On the number of diffeomorphism classes in a certain class of Riemannian manifolds. Nagoya Mathematical Journal, 97(none):173 – 192, 1985.
[Yam90] M. Yamasaki. Maps between orbifolds. Proc. Amer. Math. Soc., 109(1):223–232, 1990.

Finiteness of CAT(0)(0) group actions

Abstract.

1. Introduction

Theorem A (Finiteness).

Corollary B.

Corollary C.

Theorem D (Splitting of an Euclidean factor under ε\varepsilon-collapsed actions).

Theorem E (Renormalization).

Corollary F (Universal bound of the order of finite subgroups).

2. Preliminaries on CAT(0)(0)-spaces

2.1. Discrete isometry groups

Lemma 2.1.

Proof.

Proposition 2.2 ([CM09b, Thm.1.6 & Add.1.8], [CM09a, Cor.3.12]).

2.2. The packing condition and Margulis’ Lemma

Proposition 2.3 ([CS21, Thms. 3.1, 4.2, 4.9], [Cav22b, Lemma 3.3]).

Proposition 2.4 ([BGT11, Corollary 11.17]).

2.3. Crystallographic groups in the Euclidean space

Proposition 2.5 (Bieberbach’s Theorem).

2.4. Virtually abelian groups

Lemma 2.6.

Proof.

Proposition 2.7 ([BH13, Corollary II.7.2]).

Lemma 2.8.

Proof of Lemma 2.8.

Proposition 2.9.

Proof.

3. Systole and diastole

Theorem 3.1.

Proposition 3.2 ([CS22, Theorem A & Remark 3.5]).

Proposition 3.3.

Proof.

Proof of Theorem 3.1.

4. The splitting theorem

Theorem 4.1.

Lemma 4.2.

Proof.

Proposition 4.3.

Proof.

Lemma 4.4.

Proposition 4.5.

Proof.

Proof of Theorem 4.1.

5. The finiteness Theorems

5.1. Finiteness up to group isomorphism

Proposition 5.1.

Proof.

Proof of Theorem E.

Remark 5.2.

Proof of Theorem A.

Remark 5.3.

Proof of Corollary F.

5.2. Finiteness up to equivariant homotopy equivalence

Proposition 5.4.

Proof of Proposition 5.4..

5.3. Finiteness of nonpositively curved orbifolds

Proof of Corollary C.

References

Finiteness of CAT $(0)$ group actions

Theorem D (Splitting of an Euclidean factor under $\varepsilon$ -collapsed actions).

2. Preliminaries on CAT $(0)$ -spaces