Central limit theorem on $\operatorname{\text{CAT}}$(0) spaces with contracting isometries

Let $G$ be a discrete group acting by isometries on a proper $\operatorname{\text{CAT}}$(0) space $X$. Let $\mu$ be a probability measure on $G$, which we always assume admissible, meaning that the support of $\mu$ generates $G$ as a semigroup. Consider the sequence $\omega=(\omega_{i})_{i}$, where the $\omega_{i}^{\prime}s$ are chosen independently according to the measure $\mu$. The random walk $(Z_{n}(\omega))_{n}$ on $G$ generated by $\mu$ is then defined by $Z_{n}(\omega)=\omega_{1}\dots\omega_{n}$. Taking $o\in X$, we want to study the asymptotic behaviour of the random variables $(Z_{n}(\omega)o)_{n}$. To be more precise, we want to study limit laws of the random walk in a natural compactification of $X$. Even though these questions may be hard to solve for general metric spaces, the theory is very rich when $X$ possesses nice linear or hyperbolic-like properties. In the fundamental paper of V. Kaimanovich [Kai00], the convergence of $(Z_{n}o)_{o}$ to a point of the visual boundary is proven for groups acting geometrically on proper hyperbolic spaces and several other classes of actions. More recently this result has been extended by J. Maher and G. Tiozzo in [MT18] for groups acting by isometries on non proper hyperbolic spaces. A major difficulty in the proof of the latter result was that in the non proper setting, the completion of a hyperbolic space by its Gromov boundary might be non compact. The results of Maher and Tiozzo will be fundamental in the sequel because we will deal with hyperbolic spaces without properness assumption. In [KM99, Theorem 2.1], Karlsson and Margulis proved a first general result of convergence of the random walk on $\operatorname{\text{CAT}}$(0) spaces, under the assumption that the escape rate $\lambda=\liminf\frac{d(Z_{n}o,o)}{n}$ is positive.

In [LB22] we proved that if $G$ acts on a $\operatorname{\text{CAT}}$(0) space $X$ with rank one isometries, then the random walk $(Z_{n}(\omega))_{n}$ almost surely converges to a point of the boundary of the visual compactification $\partial_{\infty}X$. A rank one element is an axial isometry whose axes do not bound any flat half plane. We give more details on this notion in Section 2, but a rank one element must be thought of as a contracting isometry with features that typically arise in hyperbolic settings. In this context, we also prove that the escape rate (the drift) is almost surely positive: there exists $\lambda>0$ such that almost surely, $\lim_{n}\frac{d(Z_{n}o,o)}{n}=\lambda$. We review these results in Section 4. The present paper can be thought of as a continuation of [LB22], and the goal of this work is to study further limit laws of the random walk $(Z_{n}o)_{n}$, and more specifically central limit theorems for the random variables $(d(Z_{n}o,o))_{n}$.

In the case of a random product of matrices, a classical result of Furstenberg [Fur63] is the following. Take $(M_{n})$ a sequence of matrices in $\text{GL}_{n}(\mathbb{R})$, independent and identically distributed according to a probability measure $\mu$ whose support generates a noncompact subgroup of $\text{GL}_{n}(\mathbb{R})$ that does not preserve any proper linear subspace of $\mathbb{R}^{n}$. Assume that $\mu$ has finite first moment. Then there exists $\lambda>0$ such that for all $v\in\mathbb{R}^{n}-\{0\}$,

almost surely. This result can be thought of as an analogue of a law of large numbers on the random walk $(M_{n}\dots M_{1}v)_{n}$. In this context, central limit theorems and other limit laws were proven by Furstenberg-Kesten [FK60], Le Page [LP82] and Guivarc’h-Raugi [GR85]. These state that there exists $\sigma_{\mu}>0$ such that for every $v\in\mathbb{R}^{n}\setminus\{0\}$,

where $\mathcal{N}(0,\sigma^{2})$ is a centred Gaussian law on $\mathbb{R}$. We recall that the convergence in law means that for any bounded continuous function $F:\mathbb{R}\rightarrow\mathbb{R}$, one has

Those kinds of results were also obtained in negative curvature settings, for example in Gromov-hyperbolic groups [Bjö10]. However, the results stated thus far were proven under rather strong moment conditions. Typically, $\mu$ was assumed to have a finite exponential moment, that is, for which there exists $\alpha>0$ such that $\int_{G}\exp(\alpha d(o,go))d\mu(g)<\infty$.

Recently, Benoist and Quint have developed a new approach to this question and have proven central limit theorems in the linear context [BQ16b] and for hyperbolic groups [BQ16a]. They could weaken the moment condition and only assume that the measure $\mu$ has finite second moment $\int_{G}(\log\|gv\|)^{2}d\mu(g)<\infty$. Namely, if $\mu$ is such a measure on a group $G$ acting non elementarily on a proper hyperbolic space $Y$ with basepoint $o$, then there exists $\lambda>0$ such that the random variables $\frac{1}{\sqrt{n}}(d(Z_{n}(\omega)o,o)-n\lambda)$ converge in law to a non-degenerate Gaussian distribution [BQ16a, Theorem 1.1].

Using this approach, C. Horbez proved central limit theorems for mapping class groups of closed connected orientable hyperbolic surfaces and on $\operatorname{\text{Out}}(F_{N})$ [Hor18]. More recently, T. Fernós, J. Lécureux and F. Mathéus proved that if $G$ is a group acting non-elementarily on a finite-dimensional $\operatorname{\text{CAT}}$(0) cube complex, then we also have a central limit theorem for the random variables $(d(Z_{n}(\omega)o,o))_{n}$ [FLM24]. In both cases, the authors only assume a second moment condition.

The main result of this paper is to prove a similar result in the context of a group acting on a general $\operatorname{\text{CAT}}$(0) space, under the assumption that an element of the group acts as a rank one isometry. We say that the group action $G\curvearrowright X$ is non-elementary if there are no fixed points in $\overline{X}$ nor a fixed pair of points in $\partial_{\infty}{X}$.

Our strategy relies heavily on the approach developed by Benoist and Quint. To summarize, one needs to approximate the random walk by a well-chosen cocycle. Then, they give a general criterium (Theorem 5.3 below) under which this cocycle converges in law to a Gaussian distribution.

To apply this strategy, one needs to obtain good estimates on this cocycle. The general idea of this paper is then the following. In order to get a precise description of the random walk, we use a hyperbolic space that is conveniently attached to the original $\operatorname{\text{CAT}}$(0) space. As the theory of random walks in hyperbolic spaces is rich, we study the behavior of $\{Z_{n}o\}_{n}$ on this model, and then we lift this information back to the original $\operatorname{\text{CAT}}$(0) space. This strategy was implemented successfully in [Hor18] and [FLM24]:

• •

  for $\operatorname{\text{Mod}}(S)$, the hyperbolic model is the curve complex $C(S)$, and the lifting to $\mathcal{T}(S)$ is done in [Hor18, Section 3.4];

• •

  for a $\operatorname{\text{CAT}}$(0) cube complex, the hyperbolic model is the contact graph $\mathcal{C}X$, and the lifting is implemented in [FLM24, section 5].

In [PSZ22], H. Petyt, D. Spriano and A. Zallum introduced analogues of curve graphs and cubical hyperplanes for the class of $\operatorname{\text{CAT}}$(0) spaces. Using a generalized notion of hyperplane, they build a family of hyperbolic metrics $(d_{L})_{L}$ on $X$ which conserve many of the geometric features of the original $\operatorname{\text{CAT}}$(0) space. These spaces capture hyperbolic behaviours in $X$ and behave very well under the isometric action of a group. Moreover, a rank one isometry of $X$ acts on some hyperbolic model as a loxodromic isometry. Our strategy will be to chose a good hyperbolic model $X_{L}=(X,d_{L})$, and then to make use of the limit laws proven by Maher and Tiozzo in [MT18], or by Gouëzel in [Gou22]. A key fact is that there is an equivariant homeomorphic embedding of the Gromov boundary $\partial_{\text{Grom}}X_{L}$ of the hyperbolic model $X_{L}$ into the visual boundary of the $\operatorname{\text{CAT}}$(0) space [PSZ22, Theorem 7.1].

Another interesting question in the study of $(Z_{n}(\omega))_{n}$ is the proportion of steps that are "hyperbolic". In the context of random walks on hyperbolic spaces, Maher and Tiozzo show that the probability that a random walk of size $n$ is a loxodromic isometry goes to $1$ as $n$ goes to infinity [MT18, Theorem 1.4]. For a non-elementary action on an irreducible $\operatorname{\text{CAT}}$(0) cube complex, Fernós, Lécureux and Mathéus show that the proportion of steps $Z_{n}$ that are contracting goes to $1$ as $n$ goes to infinity. They use this result to show that if a group $G$ acts non-elementarily and essentially on a (possibly reducible) finite-dimensional $\operatorname{\text{CAT}}$(0) cube complex, then there exist regular elements, extending a result of Caprace and Sageev [CS11]. In our context, we also prove that "most" of the steps in the random walk are rank one. This result is not involved in the proof of Theorem 1.1, but is of independent interest.

Using the curtain models from [PSZ22], such a result is actually straightforward. We emphasize the idea that a systematic approach of dynamics on $\operatorname{\text{CAT}}$(0) spaces using these hyperbolic models can prove fruitful, especially when quantitative estimates are required. Indeed, curtain models also benefit from their combinatorial structure. In this paper, we exploit this richness in several ways, especially in the main geometric lemma 6.1. In [LB23, Section 5], the author develops these connections in order to study other limit laws on general Hadamard spaces.

A different approach for the study of such limit laws was implemented in [MS20], where the authors prove central limit theorems on acylindrically hyperbolic groups. Their strategy relies on a control of deviation inequalities, which encapsulate the way the random walk progresses in an “almost aligned” way, hence their approach apply to possibly non-proper spaces. While there is a slight overlap with the results stated here (especially [MS20, Theorem 13.4]), Mathieu and Sisto study random walks on acylindrically hyperbolic groups with a word metric. This situation does not immediately apply here. Indeed, in our main theorem, the pull-back metric induced on G by an orbit map need not be quasi-isometric to a word metric, and in fact need not even be proper. Also, their assumptions on the measure $\mu$ are much more restrictive: they assume that $\mu$ has finite exponential moment. In particular, their assumption is not optimal, while it is the case here. Last, the techniques involved are completely different: their approach has a “local” flavor, whereas here we use boundary theory and compactifications.

While we were working on this project, Inhyeok Choi released a paper in which he states central limit theorems along with other limit laws in $\operatorname{\text{CAT}}$(0) spaces, Teichmüller spaces and outer spaces [Cho22]. One of the main assumptions is still the presence of a pair of independent contracting isometries in the group, but the methods and the proofs are different. Indeed, Choi uses a pivotal technique introduced by Boulanger, Mathieu and Sisto in [MS20] and [BMSS23] and further developed by Gouëzel in [Gou22]. These techniques have a “local flavor”, while our paper relies on boundary theory, and uses hyperbolic models that depend on specific features of $\operatorname{\text{CAT}}$(0) spaces. We think this approach is natural from a geometric point of view, and we believe that the interplay between $\operatorname{\text{CAT}}$(0) spaces and their underlying hyperbolic models will be useful in the study of still open questions about limit laws. In any case, it is always interesting to have different strategies and techniques for studying radom walks and limit laws.

The essential assumption in these results is the presence of contracting elements for the action of $G$ on the $\operatorname{\text{CAT}}$(0) space $X$. In [LB23, Chapter 5], the author proves that actually, all the results presented here hold in the more general context of a Hadamard space, i.e. a separable and complete $\operatorname{\text{CAT}}$(0) space, removing the properness assumption on $X$. Notice that the boundary of a general Hadamard spaces may be no longer compact, and that the embedding of the boundaries $\partial_{\text{Grom}}X_{L}\hookrightarrow\partial_{\infty}X$ is only stated for proper $\operatorname{\text{CAT}}$(0) spaces in [PSZ22]. In [LB23, Theorem 5.3.5], we prove that this embedding actually holds in this more general setting. Once this is done, the general strategy for proving the central limit theorem 1.1 is similar, although with some additional technical difficulties. We refer to the manuscript [LB23, Chapter 5] for details.

We believe our approach can be of use in order to determine if the boundary $\partial_{\infty}X$ endowed with the hitting measure is actually the Poisson boundary of $(G,\mu)$, extending a result of Karlsson and Margulis for cocompact actions [KM99, Corollary 6.2].

Moreover, it seems natural to use these hyperbolic spaces to prove that if $\mu$ has finite first moment, then limit points of the random walk almost surely belong to the sublinear Morse boundary constructed by Qing and Rafi in [QR22]. Note that this question is linked to the previous one, because it is believed that the sublinear Morse boundary is often a good candidate for the Poisson boundary, especially for finitely supported measures, see for example [QR22, Theorem F] and [QRT20, Theorem B]. In both cases, the use of hyperbolic models seems useful because of precision of estimates that can be derived from the combinatorial structure of these spaces.

In Section 2, we review basic definitions about random walks, rank one isometries and explain our setting. In Section 3, we explicit the construction and properties of the hyperbolic models $(X,d_{L})$, and give various geometric lemmas that will be useful afterwards. Section 4 is dedicated to presenting the works of Maher and Tiozzo in [MT18] and of Gouëzel in [Gou22], and the first results in proper $\operatorname{\text{CAT}}(0)$ spaces that were found in [LB22]. We explain the strategy developed by Benoist and Quint in Section 5, and give the proof of our main Theorem in Section 6.

The goal of this section is to briefly present some ideas of [PSZ22], in which the authors build a way of attaching a family of hyperbolic metric spaces $X_{L}=(X,d_{L})_{L}$ to a proper $\operatorname{\text{CAT}}$(0) space. What is interesting about these spaces is that they convey much of the geometry of the original space, especially at infinity, and they behave very well under isometric actions. More specifically, rank one isometries will act on some well-chosen spaces as loxodromic isometries. This construction can be understood as the analogue (and generalization) of the curve graphs that exist in the context of $\operatorname{\text{CAT}}$(0) cube complexes, see [Hag14] and [Gen19].

We will often denote a curtain by the letter $h$, even though one must keep in mind that $h=h_{\gamma,t}$ is characterized by a given geodesic $\gamma:I\rightarrow X$ and a point $t\in I$ (which defines a unique pole $P\subseteq\gamma$). Sometimes, we may also write $h=h_{\gamma,P}$ to emphasize on the pole $P$.

Curtains can fail to be convex: if $x,y\in h^{-}$, it may happen that there exists $z\in[x,y]\cap h^{+}$, see [PSZ22, Remark 3.4]. Nonetheless, we have a weaker notion of convexity that the authors call star convexity:

One can check that this definition gives a metric. If $h$ is a curtain, we have seen that $d(h^{-},h^{+})=1$, hence for any $x,y\in X$, $d_{\infty}(x,y)\leq\lceil d(x,y)\rceil$. Conversely, it turns out that $d$ and $d_{\infty}$ may differ by at most 1, as shown by the following lemma.

We are now ready to refine the notion of separation in order to capture only some of the hyperbolic features of the space.

We say that a chain $c$ of curtains meets a curtain $h$ if every single curtain $h_{i}\in c$ intersects $h$.

The following geometric Lemma is a key ingredient for the proof of Theorem 3.10, and will be used several times in the sequel. It means that $L$-separation induces good Morse properties. The picture one has to keep in mind is given by Figure 1.

The next Lemma states that if there is a $L$-chain separating two points $x$ and $y$, we can find a (smaller) $L$-chain of curtains separating those, which is dual to the geodesic $[x,y]$ and whose size can be controlled. It will prove useful later on, especially when we want to use Lemma 3.7.

We are now ready to define a family of metrics using $L$-separation.

It turns out that for every $L$, $d_{L}$ gives a metric on $X$ [PSZ22, Lemma 2.17]. We will denote by $X_{L}=(X,d_{L})$ the resulting metric space. With this definition in hand, Petyt, Spriano and Zalloum prove that the metric spaces $(X,d_{L})$ are hyperbolic.

We will then call $(X,d_{L})$ a hyperbolic model for the $\operatorname{\text{CAT}}$(0) space $X$. Another useful fact about these spaces is that they behave well under isometries with "hyperbolic-like" properties.

Another piece of information brought by this construction is the relation between the Gromov boundaries $\partial X_{L}$ of the hyperbolic models $X_{L}=(X,d_{L})$ and the visual boundary of the original $\operatorname{\text{CAT}}$(0) space $(X,d)$.

As a consequence of Lemma 3.7 and Lemma 3.8, if two geodesic rays with the same starting point cross an infinite $L$-chain $c$, then they are asymptotic, and hence equal.

Given $o\in X$, we define $\mathcal{B}_{L}$ as the subspace of $\partial_{\infty}X$ consisting of all geodesic rays $\gamma:[0,\infty)\rightarrow X$ starting from $o$ and such that there exists an infinite $L$-chain crossed by $\gamma$. In the case of the contact graph associated to a $\operatorname{\text{CAT}}$(0) cube complex $X$, we had the existence of an $\operatorname{Isom}(X)$-equivariant embedding of the boundary of the contact graph into the Roller boundary $\partial_{\mathcal{R}}X$. The following result is the analogue in the context of $\operatorname{\text{CAT}}$(0) spaces.

Recall that the support of a Borel measure $m$ on a topological space $Y$ is the smallest closed set $C$ such that $m(Y\setminus C)=0$. In other words $y\in\operatorname{\text{supp}}(m)$ if and only if for all $U$ open containing $y$, $m(U)>0$.

In order to use the results concerning random walks in hyperbolic spaces, we must show that the action of a group $G$ on a proper $\operatorname{\text{CAT}}$(0) space with rank one isometries induces a non-elementary action on some hyperbolic model $(X,d_{L})$.

The results of Section 3 allow us to read some information about the random walk in the hyperbolic models $X_{L}=(X,d_{L})$, and then translate this information back to the original $\operatorname{\text{CAT}}$(0) space. As the theory of random walks on hyperbolic spaces is well-studied, one may hope that this process is fruitful.

In order to prove our main result, we use a strategy that is largely inspired by the works of Benoist and Quint on linear spaces and hyperbolic spaces, see [BQ16a] and [BQ16b]. They developed a method for proving central limit theorems for cocycles, relying on results due to Brown in the case of martingales [Bro71].