2.1 Graph structures for countable groups

Given a countable group $G$, we define a graph structure on $G$ as a triple $(\Gamma, v_0, \mathrm {ev})$, where $\Gamma$ is a finite, directed graph, $v_0$ is a vertex of $\Gamma$ which we call its initial vertex, and $\mathrm {ev} \colon E(\Gamma ) \to G$ is a map that labels the edges of $\Gamma$ with group elements. Given this data, we extend the map $\mathrm {ev}$ by defining for each finite path $g = g_1 \cdots g_n$ the group element $\mathrm {ev}(g) = \mathrm {ev}(g_1) \cdots \mathrm {ev}(g_n)$. We remark that here and throughout, a path in $\Gamma$ will always mean a directed path. To simplify notation, we will use $\overline {g} = \mathrm {ev}(g)$ to denote the group element associated to the path $g$. We denote by $\Vert g \Vert$ the length of the path $g$. The graph structure is bounded if there is a uniform bound on the number of paths in $\Gamma$ mapping to any single element of $G$.

For a graph structure $\Gamma$, we define $\Omega$ to be the set of all infinite paths starting at any vertex of $\Gamma$ and $\sigma \colon \Omega \to \Omega$ to be the shift map. Given a path $\omega = (g_1, \ldots, g_n, \ldots )$, we denote by $w_n := g_1 \cdots g_n$ its prefix of length $n$. The set of all finite paths starting at any vertex of $\Gamma$ will be denoted by $\Omega ^{*}$.

We define two vertices $v_i, v_j$ to be equivalent if there is a path from $v_i$ to $v_j$ and a path from $v_j$ to $v_i$, and the components of $\Gamma$ as the equivalence classes for this relation.

We will denote by $M$ the transition matrix for $\Gamma$. By Perron–Frobenius, $M$ has a real eigenvalue of largest modulus, which we will denote by $\lambda$. Moreover, such a matrix is almost semisimple if for any eigenvalue of maximal modulus, its geometric and algebraic multiplicity agree. Furthermore, such a matrix is semisimple if its only eigenvalue of maximal modulus is real positive. We call a graph structure (almost) semisimple if its associated transition matrix is.

Let $\Gamma$ be almost semisimple, and let $\lambda$ be the leading eigenvalue of $M$. Then we define a vertex $v$ to be of large growth if

\begin{align*} \lim_{n \to \infty} \frac{1}{n} \log \# \{ \text{paths of length }n\text{ starting at }v \} = \lambda \end{align*}

and of small growth otherwise (in which case, the limit above is less than $\lambda$). Furthermore, a component $C$ is maximal if

\begin{align*} \lim_{n \to \infty} \frac{1}{n} \log \# \{ \text{paths of length }n\text{ inside }C \} = \lambda. \end{align*}

As discussed in [Reference Gekhtman, Taylor and TiozzoGTT20], the global structure of $\Gamma$ is as follows: there is no path between maximal components, and vertices of large growth are precisely the ones which have a path to a maximal component.

Given a vertex $v$, we denote by $\Gamma _v$ the loop semigroup of $v$, that is, the set of all finite paths from $v$ to itself. This is a semigroup under concatenation, and all its elements lie entirely in the component of $v$. We denote by $\overline {\Gamma }_v$ the image of $\Gamma _v$ in $G$ under the evaluation map.

Definition 2.1 (Thick graph structure)

A graph structure $\Gamma$ is thick if, for any vertex $v$ in a maximal component, there exists a finite set $B \subseteq G$ such that

\begin{align*} G = B \cdot \overline{\Gamma}_v \cdot B, \end{align*}

where the equality is in the group $G$.

In what follows, we often make the evaluation map implicit in our notation. In particular, if $G$ acts on a metric space $(X, d)$, $o \in X$ is a base point, and $g$ is a finite path in $\Gamma$, we will often write $go$ to mean the point $\overline {g}o \in X$.

2.1 Geodesic combings

For particular applications, it is also useful to define the notion of a geodesic graph structure. A graph structure $\Gamma$ is geodesic if the length $\Vert g \Vert$ of any path $g$ is equal the word length of $\overline {g}$ in the subgroup generated by the edge labels, using edge labels as the (finite) generating set. A geodesic graph structure is called a geodesic combing if, in addition, there is a directed path from $v_0$ to any other vertex of $\Gamma$ and the evaluation map is a bijection from the set of finite paths starting at $v_0$ to the set of elements of $G$. Note that a geodesic combing is automatically a bounded graph structure. We say that $\Gamma$ is a geodesic combing associated to a finite generating set $S$ if, up to adding inverses, $S$ is the set of edge labels for the graph structure. In this case, $\Vert g \Vert$ is equal to the word length of $\overline {g}$ with respect to $S$.

We emphasize that although the geodesic condition is used in the applications of our main theorem (as in Theorems 1.1–1.6), it is not required for the most general results; see Theorems 7.3 and 7.4. There we use the following more general notion.

Definition 2.2 (Coherent graph structure)

A graph structure $\Gamma$ for $G$ is coherent if it is bounded and if for any finite set $B \subseteq G$ there exists a constant $\mathcal {B} \ge 0$ such that if $g$ and $h$ are finite length paths in $\Gamma$, and $\overline {g} = b_1 \overline {h} b_2$ in $G$, then

\begin{align*} | \Vert g \Vert - \Vert h \Vert | \leq \mathcal{B}. \end{align*}

The idea of coherence is simple; informally, if two group elements are close in $G$ then a coherent graph structure $\Gamma$ codes them with paths that are roughly the same length, and the coding is uniformly bounded-to-one.

Our first lemma summarizes some properties of thick, coherent graph structures that we will need in the sequel.

Lemma 2.3 A thick, coherent graph structure $\Gamma$ is almost semisimple.

Moreover, if $G \curvearrowright X$ is a nonelementary action on a hyperbolic space, then the actions of both semigroups $\Gamma _v$ and $\Gamma _v^{-1}$ on $X$ are also nonelementary, for each vertex $v$ contained in a maximal component.

Proof. Suppose the graph structure $\Gamma$ is thick and coherent. Since a coherent graph structure is bounded, there is a $\mathfrak b \ge 0$ so that the evaluation map $\mathrm {ev} \colon \Omega ^{*} \to G$ is at most $\mathfrak b$-to-one. Let $B\subset G$ be the finite set given by thickness of $\Gamma$ (Definition 2.1) and let $\mathcal {B}$ be the resulting constant from Definition 2.2. By thickness and coherence we have

\begin{align*} \overline{S_n} \subseteq \bigcup_{b_1, b_2 \in B} \bigcup_{|k| \leq \mathcal{B}} b_1 (\overline{\Gamma_v \cap S_{n + k}}) b_2, \end{align*}

where $S_n \subset \Omega ^{*}$ is the set of all length $n$ paths starting at any vertex. Then the definition of $\mathfrak b$ implies that

\begin{align*} \# S_n \leq \mathfrak{b} |B|^{2} \sum_{|k| \leq \mathcal{B}} \# (\Gamma_v \cap S_{n+ k}). \end{align*}

This, in turn, is bounded by a constant times $\lambda ^{n}$ since it is no more than the growth of paths in the maximal component of $\Gamma$ containing $v$, and $k$ is uniformly bounded. On the other hand, if the transition matrix $M$ of $\Gamma$ is not almost semisimple, then $M$ has a Jordan block for an eigenvalue of modulus $\lambda$ of size $k\ge 2$ (see [Reference Gekhtman, Taylor and TiozzoGTT18, § 2]). In particular, the growth of paths in $\Gamma$ is at least a constant times $n^{k-1}\lambda ^{n}$. We conclude that $\Gamma$ is almost semisimple.

The statement that the action of $\Gamma _v$ (and hence $\Gamma _v^{-1}$) on $X$ is nonelementary is proven in [Reference Gekhtman, Taylor and TiozzoGTT20, Proposition 6.3].

The following lemma is immediate from the definitions.

Lemma 2.4 A geodesic combing for $G$ is coherent.

Other sources of (not necessarily geodesic) coherent graph structures come from biautomatic groups, in the sense of [Reference MosherMos97] or [Reference Epstein, Paterson, Cannon, Holt, Levy and ThurstonEPC⁺92, Lemma 2.5.5].

2.2 Cocycles and horofunctions

Let $(X, d)$ be a metric space, and let $o \in X$ be a base point. Given $z \in X$, we define the Busemann function $\rho _z : X \to \mathbb {R}$ as

\begin{align*} \rho_z(x) := d(x, z) - d(o, z). \end{align*}

Thus, setting

\begin{align*} \Phi(z) := \rho_z \end{align*}

defines a map

\begin{align*} \Phi : X \to \text{Lip}^{1}_o(X) \end{align*}

where $\text {Lip}^{1}_o(X)$ is the space of $1$-Lipschitz functions on $X$ which vanish at $o$.

We define the horofunction compactification $\overline {X}^{h}$ as the closure of $\Phi (X)$ in $\text {Lip}^{1}_o(X)$, with respect to the topology of pointwise convergence. Elements of $\overline {X}^{h}$ will be called horofunctions. We denote by $\overline {X}_\infty ^{h}$ the space of infinite horofunctions, that is, the set of $h \in \overline {X}^{h}$ such that $\inf _{x \in X} h(x) = - \infty$.

For any $\xi \in \overline {X}^{h}$, the Busemann cocycle is defined as

\begin{align*} \beta_{\xi}(x, y) &:= \lim_{z_n \to \xi} [ d(y, z_n) - d(x, z_n)] \\ &= h_\xi(y) - h_\xi(x), \end{align*}

where $h_\xi$ is the horofunction associated to $\xi$. This has the usual cocycle property $\beta _{\xi }(x, z) =\beta _{\xi }(x, y) +\beta _{\xi }(y, z)$.

Remark 2.5 Benoist and Quint [Reference Benoist and QuintBQ16] and Horbez [Reference HorbezHor18] define $B \colon G \times \overline {X}^{h} \to \mathbb {R}$ by

\begin{align*} B(g, \xi) = h_\xi(g^{-1} o). \end{align*}

To compare their definition with ours:

\begin{align*} B(g, \xi) = h_\xi(g^{-1} o) = \lim_{z_n \to \xi} [ d(g^{-1}o, z_n) - d(o, z_n) ] = \beta_\xi(o, g^{-1} o). \end{align*}

3.1 Central limit theorems for cocycles

Recall that a cocycle is a function $\sigma \colon G \times \mathcal {M} \to \mathbb {R}$ such that

\begin{align*} \sigma(gh, x) = \sigma(g, h x) + \sigma(h, x),\quad \forall\ g, h \in G,\enspace \forall\ x \in \mathcal{M}. \end{align*}

A cocycle $\sigma \colon G \times \mathcal {M} \to \mathbb {R}$ has constant drift $\lambda$ if there exists $\lambda \in \mathbb {R}$ such that

\begin{align*} \int_G \sigma(g, x) \, d\mu(g) = \lambda \end{align*}

for any $x \in \mathcal {M}$. A cocycle $\sigma \colon G \times \mathcal {M} \to \mathbb {R}$ is centerable if it can be written as

\begin{align*} \sigma(g, x) = \sigma_0(g, x) + \psi(x) - \psi(g \cdot x), \end{align*}

where $\sigma _0$ is a cocycle with constant drift and where $\psi \colon \mathcal {M} \to \mathbb {R}$ is a bounded, measurable function. In this case, we say that $\sigma _0$ is the centering of $\sigma$; note that $\lambda = \int _{G \times \mathcal {M}} \sigma (g, x) \, d\mu (g) \,d \nu (x)$ for any $\mu$-stationary $\nu$. We say that the cocycle $\sigma$ has finite second moment with respect to a measure $\mu$ on $G$ if

\begin{align*} \int_G \sup_{x\in \mathcal{M}} | \sigma(g,x)|^{2} \, d\mu(g)< +\infty. \end{align*}

We now use the following CLT for centerable cocycles; as remarked in [Reference HorbezHor18, Remark 1.7], the proof is exactly the same as the proof of [Reference Benoist and QuintBQ16, Theorem 4.7].

Theorem 3.1 (Central limit theorem for cocycles)

Let $G$ be a discrete group, $\mathcal {M}$ be a compact metrizable $G$-space and $\mu$ a probability measure on $G$. Let $\nu$ be a $\mu$-ergodic, $\mu$-stationary probability measure on $\mathcal {M}$, and let $\mathcal {M}_0$ be a $G$-invariant subset of $\mathcal {M}$ of full $\nu$-measure. Let $\sigma \colon G \times \mathcal {M}_0 \to \mathbb {R}$ be a centerable cocycle with drift $\lambda$ and finite second moment. Then there exist $\sigma \geq 0$ such that for any continuous $F \colon \mathbb {R} \to \mathbb {R}$ with compact support, we have for $\nu$-almost every (a.e.) $x \in \mathcal {M}$,

\begin{align*} \lim_{n \to \infty} \int_G F\bigg(\frac{\sigma(g, x) - \lambda n}{\sqrt{n}}\bigg) d \mu^{*n}(g) = \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t). \end{align*}

We now apply this result to the loop semigroup. Let $F_v$ be the group freely generated by the prime loops in $\Gamma _v$.

Let $N\colon \Gamma _v \to \mathbb {Z}$ be the semigroup homomorphism $N(g) := - \Vert g \Vert$, where $\Vert g \Vert$ is the length in $\Gamma$ of the loop $g$. There is a natural inclusion $\Gamma _v \to F_v$ as a subsemigroup and we can extend the semigroup homomorphism above to a group homomorphism $N\colon F_v \to \mathbb {Z}$. Moreover, we also extend the natural semigroup homomorphism $\Gamma _v \to G$, induced by evaluation, to a group homomorphism $e \colon F_v \to G$. Now, using the homomorphism $e \colon F_v \to G$, the free group $F_v$ has a nonelementary action on $X$, and, moreover, $\mu _v^{*n}$ is supported on $\Gamma _v \subseteq F_v$ for all $n\ge 1$.

Finally, for some $\ell \in \mathbb {R}$ to be specified below, we define $\eta \colon F_v \times \overline X^{h} \to \mathbb {R}$ as

\begin{align*} \eta(g, \xi) := \beta_{\xi}(o, g^{-1} o) - \ell N(g). \end{align*}

Proof. We have

\begin{align*} \eta(gh, \xi) &= \beta_\xi(o, h^{-1} g^{-1} o) - \ell N(gh) \\ &= \beta_\xi(o, h^{-1} o) + \beta_{\xi}(h^{-1} o, h^{-1} g^{-1} o) - \ell N(g) - \ell N(h) \\ &= \beta_\xi(o, h^{-1} o) + \beta_{h \xi}(o, g^{-1} o) - \ell N(g) - \ell N(h) \\ &= \eta(h, \xi) + \eta(g, h\xi), \end{align*}

hence $\eta$ is a cocycle. Moreover, by [Reference HorbezHor18, Proposition 1.5], using [Reference HorbezHor18, Corollary 2.7] and [Reference HorbezHor18, Proposition 2.8], the cocycle $B(g, \xi ) = \beta _\xi (o, g^{-1}o)$ is centerable on $F_v \times \overline {X}_\infty ^{h}$. Then, since $\eta (g, \xi ) - B(g, \xi ) = \ell N(g)$ is a homomorphism and depends only on $g$, we have that $\eta (g, \xi )$ is also centerable on $F_v \times \overline {X}_\infty ^{h}$.

Thus, we obtain the following consequence of Theorem 3.1.

Corollary 3.3 Let $\Gamma$ be a thick structure, let $v$ be a vertex in a maximal component of $\Gamma$. Suppose that the first return measure $\mu _v$ is nondegenerate, and let $\nu _v$ be a $\check {\mu }_v$-ergodic, $\check {\mu }_v$-stationary measure on $\overline {X}^{h}$. Then there exist $\ell, \sigma \ge 0$ such that for any continuous $F \colon \mathbb {R} \to \mathbb {R}$ with compact support, we have for $\nu _v$-a.e. $\xi$,

\begin{align*} \lim_{n \to \infty} \int_G F \bigg(\frac{\beta_\xi(o, g o) - \ell \Vert g \Vert}{\sqrt{n}}\bigg) d \mu_v^{*n}(g) = \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t). \end{align*}

Proof. We apply Theorem 3.1 to the measure $\check {\mu }_v$, supported on $\Gamma _v^{-1}$, where $\ell$ is chosen so that $\lambda = \int _{F_v \times \overline X^{h}} \eta (g, \xi ) \, d\check {\mu }_v(g) \,d\nu _v(\xi ) = 0$. Note that, by [Reference Maher and TiozzoMT18, Proposition 4.4] and the fact that $\Gamma _v^{-1}$ is nonelementary (Lemma 2.3), we have $\nu _v(\overline {X}^{h}_\infty ) = 1$. Moreover, for any $g \in \Gamma _v$ we have

\begin{align*} \eta(g^{-1}, \xi) = \beta_{\xi}(o, g o) - \ell \Vert g \Vert. \end{align*}

3.2 Skew products and invariance on the loop semigroup

Let $\mathcal {M}$ be a compact metric space with a continuous $G$-action. We define the skew product $T\colon \Omega \times \mathcal {M} \to \Omega \times \mathcal {M}$ as

\begin{align*} T(\omega, \xi) := (\sigma(\omega), g_1^{-1} \xi) \end{align*}

where $\omega = (g_1, g_2, \ldots )$.

A graph structure $\Gamma$ is primitive if its associated transition matrix $M$ is primitive, that is, has a positive power. Now let $\Gamma$ be a primitive graph structure, let $v$ be a vertex of $\Gamma$, let $\Gamma _v$ be the loop semigroup, and let $\mu _v$ be the first return measure.

Finally, let $\Omega _v = (\Gamma _v)^{\mathbb{N}}$ with shift map $\sigma _v$. To highlight the difference, we denote the elements of $\Gamma _v^{\mathbb{N}}$ by $(l_1, l_2, \ldots )$, since each element of the sequence is a loop, while the elements of $\Omega$ will be denoted by $\omega = (g_1, g_2, \ldots )$, since its elements are edges. Let us define the map $T_v : \Omega _v \times \mathcal {M} \to \Omega _v \times \mathcal {M}$ as

\begin{align*} T_v(\omega, \xi) = (\sigma_v (\omega), l_1^{-1} \xi). \end{align*}

Proof. Fix $C \subset \Omega _v$ measurable and let $C_l \subset \Omega _v$ be the subset consisting of sequences beginning with $l \in \Gamma _v$ such that $\sigma _v (C_l) = C$. Then, for any $A \subset \mathcal {M}$ measurable,

\begin{align*} T_v^{-1} (C \times A) = \bigcup_l C_l \times l A. \end{align*}

Since

\begin{align*} \mu_v^{\mathbb{N}} \otimes \nu \bigg(\bigcup_l C_l \times l A\bigg) &= \mu(C) \sum_l \mu(l) \nu(lA) \\ &= \mu(C) \sum_l \check \mu(l) l_*\nu(A), \end{align*}

the lemma follows.

Lemma 3.6 Consider the function $f\colon \Omega \times \overline {X}^{h} \to \mathbb {R}$ defined as

\begin{align*} f(\omega, \xi) := \beta_\xi(o, g_1 o). \end{align*}

Then for any $n$ we have

(1)\begin{equation} \sum_{j = 0}^{n-1} f(T^{j}(\omega, \xi)) = \beta_{\xi}(o, w_n o). \end{equation}

Proof. The cocycle property implies

\begin{align*} \beta_{\xi} (o, w_n o) = \sum_{j =0}^{n-1} \beta_{\xi} (w_j o, w_{j+1} o) = \sum_{j = 0}^{n-1} \beta_{w_j^{-1} \xi}(o, g_{j+1} o) \end{align*}

for any $\xi \in \overline {X}^{h}$. Moreover, by definition and $G$-equivariance we have

\begin{align*} f(T^{j}(\omega, \xi)) = \beta_{w_j^{-1} \xi}(o, g_{j+1} o) \end{align*}

and the claim follows.

An analogous statement holds by replacing $T, \Omega$ by $T_v, \Omega _v$.

4.1 Suspension flows

Let $S \colon (\mathcal {X}, \lambda ) \to (\mathcal {X}, \lambda )$ be a measure-preserving dynamical system, and let $r \colon \mathcal {X} \to \mathbb {N}$ be a measurable, integrable function, which we call the roof function. Then the discrete suspension flow of $S$ with roof function $r$ is the dynamical system given by the map $\widehat {S} : \widehat {\mathcal {X}} \to \widehat {\mathcal {X}}$ where

\begin{align*} \widehat{\mathcal{X}} := \{ (x, n) \in \mathcal{X} \times \mathbb{N} : 0 \leq n \leq r(x) - 1\} \end{align*}

with measure $\widehat {\lambda } := ({1}/{\overline {r}}) (\lambda \otimes \delta )$, where $\delta$ is the counting measure on $\mathbb {N}$ and $\overline {r} := \int _{\mathcal {X}} r \, d\lambda$. Then the map $\widehat {S}$ is defined as

\[ \widehat{S}(x, n) = \left\{\begin{array}{@{}ll} (x, n+1) & \text{if }n \leq r(x) - 2, \\ (S(x), 0) & \text{if }n = r(x) - 1. \end{array}\right. \]

Since in this case the system has discrete time, the above construction is also called a Kakutani skyscraper.

We now state the main theorem of Melbourne and Törok [Reference Melbourne and TörökMT04, Theorem 1.1].

4.2 Invariant measure on the suspended space

For any $\omega \in \Omega _v$, let $r(\omega )$ be the length in $\Gamma$ of $l_1(\omega )$. This is the first return time for the loop determined by $\omega$. Let us define the suspension of the skew product

\[ \Omega^{(s)} := \{ (\omega, k, \xi) \in \Omega_v \times \mathbb{N} \times \mathcal{M} : 0 \leq k \leq r(\omega) - 1\} \]

and

\begin{gather*} \widehat{T}(\omega, k, \xi) = \left\{ \begin{array}{@{}ll} (\omega, k+1, \xi) & \text{if }k \leq r(\omega) - 2,\\ (\sigma_v(\omega), 0, l^{-1}_1 \xi) & \text{if } k = r(\omega) - 1. \end{array}\right. \end{gather*}

Let us now denote $R := \int _{\Gamma _v} \Vert g \Vert \, d \mu _v(g) = \int r(\omega ) \, d \mathbb {P}_v(\omega )$ and define the probability measure $\nu ^{(s)} := ({1}/{R})(\mu _v^{\mathbb{N}} \otimes \delta \otimes \nu _v)$ on $\Omega ^{(s)}$.

Proof. It suffices to check invariance of the measure using cylinder sets $C_{l_1, \ldots, l_n}$ consisting of loops beginning with $l_1\cdots l_n$. We have

\[ \widehat{T}^{-1}(C_{l_1, \ldots, l_n} \times \{ k \} \times A) = \left\{ \begin{array}{@{}ll} C_{l_1, \ldots, l_n} \times \{ k-1 \} \times A & \text{if }k > 0,\\ \displaystyle\bigsqcup_{l \in P_v} C_{l, l_1, \ldots, l_n} \times \{ \Vert l \Vert -1 \} \times l A & \text{if }k = 0,\\ \end{array}\right. \]

where $P_v \subseteq \Gamma _v$ is the set of prime loops. Hence, in the first case, the equality

\begin{align*} \nu^{(s)}(\widehat{T}^{-1}(C_{l_1, \ldots, l_n} \times \{ k \} \times A)) = \nu^{(s)}(C_{l_1, \ldots, l_n} \times \{ k \} \times A) \end{align*}

is obvious. In the second case,

\begin{align*} \nu^{(s)}(\widehat{T}^{-1}(C_{l_1, \ldots, l_n} \times \{ k \} \times A)) &= \frac{1}{R} \sum_{l \in P_v} \mu_v(l) \mu_v(l_1)\cdots \mu_v(l_n) \nu_v(lA) \\ &= \frac{1}{R} \mu_v(l_1)\cdots \mu_v(l_n) \sum_{l \in P_v} \mu_v(l) \nu_v(lA) \\ &= \frac{1}{R} \mu_v(l_1)\cdots \mu_v(l_n) \nu_v(A) \\ &= \nu^{(s)}(C_{l_1, \ldots, l_n} \times \{ k \} \times A), \end{align*}

hence $\nu ^{(s)}$ is $\widehat {T}$-invariant. Moreover, the suspension of an ergodic measure is ergodic (see, for example, [Reference SarigSar20, Proposition 1.11]).

4.3 Pushforward of the $\widehat {T}$-invariant measure to $\Omega \times \mathcal {M}$

Recall that $\Omega$ is the space of all infinite sample paths in $\Gamma$ starting at any vertex. Let us define the projection $\pi : \Omega ^{(s)} \to \Omega \times \mathcal {M}$ as

\begin{align*} \pi(\omega, k, \xi) = (\sigma^{k}(\omega), (g_1\cdots g_k)^{-1} \xi) \end{align*}

and recall that the skew product $T: \Omega \times \mathcal {M} \to \Omega \times \mathcal {M}$ is

\begin{align*} T(\omega, \xi) := (\sigma(\omega), g_1^{-1} \xi). \end{align*}

Lemma 4.4 The following diagram commutes.

As a consequence, in the hypotheses of the previous lemmas, the measure $\overline {\nu } := \pi _\star \nu ^{(s)}$ is $T$-invariant and ergodic.

Proof. We show that the horizontal arrow is equivariant for the shifts. This follows from the fact that if we write $l_1(\omega )$ for the first return loop of $\omega$ then $l_1(\omega ) = g_1(\omega )\ldots g_{r(\omega )}(\omega )$. Hence,

\begin{align*} \pi \circ \widehat T (\omega, r(\omega) -1, \xi) &= \pi ((\sigma_v(\omega),0, l_1^{-1}\xi))\\ &= (\sigma^{r(\omega)}(\omega), (g_1\cdots g_{r(\omega)})^{-1}\xi), \end{align*}

which is equal to $T \circ \pi ((\omega, r(\omega ) -1, \xi ))$. The other cases being trivial, this proves the first statement.

Finally, since $\overline \nu$ is the pushforward of an ergodic measure, it is ergodic.

4.4 Return times and invariant measures for the Markov chain

Recall that in the previous section we produced a measure $\nu _v$ on $\mathcal {M}$ which is $\check {\mu }_v$-stationary and such that the product measure $\mu _v^{\mathbb{N}} \otimes \nu _v$ is $T_v$-invariant and ergodic. Then, by lifting it to the suspension and pushing it forward to $\Omega \times \mathcal {M}$, we have an ergodic, $T$-invariant measure $\overline {\nu }$ on $\Omega \times \mathcal {M}$.

Now, for any vertex $w$ other than $v$ we define the measure $\nu _w$ on $\mathcal {M}$ as

(2)\begin{equation} \nu_w(A) = \sum_{\gamma \in \Gamma_{v, w}} \mu(\gamma) \nu_v(\gamma A) \end{equation}

where the sum is over the set $\Gamma _{v, w}$ of all paths $\gamma$ from $v$ to $w$ which do not pass through $v$ in their middle, and $\mu (\gamma )$ is the product of the measures of the edges of $\gamma$. Recall also we denote by $\mathbb {P}_w$ the Markov measure on the space of infinite sample paths starting at $w$.

Lemma 4.5 We have

\begin{align*} \overline{\nu} := \frac{1}{R} \sum_w \mathbb{P}_w \otimes \nu_w. \end{align*}

Proof. Let $w$ be a vertex, and let $g_1, g_2, \ldots, g_n$ be a finite path starting from $w$. We have for any measurable $A \subseteq \mathcal {M}$

\[ \pi^{-1}(C_{g_1, \ldots, g_n} \times A) = \left\{ \begin{array}{@{}ll} C_{g_1, \ldots, g_n} \times \{0 \} \times A & \text{if }w = v, \\ \displaystyle\bigsqcup_{\gamma \in \Gamma_{v, w}} C_{\gamma, g_1, \ldots, g_n} \times \{|\gamma|\} \times \gamma A & \text{if }w \neq v, \\ \end{array}\right. \]

where the union is over the set $\Gamma _{v, w}$ of all paths $\gamma$ from $v$ to the initial vertex $w$ of $g_1$ which do not pass through $v$ in their middle. Thus we have

\begin{align*} \overline{\nu}(C_{g_1, \ldots, g_n} \times A) &= \frac{1}{R} \sum_{\gamma \in \Gamma_{v, w}} \mu(\gamma) \mu(g_1) \cdots \mu(g_n) \nu_v(\gamma A) \\ & = \frac{1}{R} \mu(g_1) \cdots \mu(g_n) \nu_w( A) \\ & = \frac{1}{R} \mathbb{P}_w(C_{g_1, \ldots, g_n}) \nu_w(A) \end{align*}

which proves the claim, since both measures agree on all rectangles.

Recall that $R = \int r(\omega ) \, d\mathbb {P}_v(\omega )$, and set $n_w = \nu _w(\mathcal {M})$. Here we show the following lemma.

Note that if we replace $\Gamma$ with the graph $\overline \Gamma$ obtained by reversing the direction of each edge, then the transition matrix for $\overline \Gamma$ is $M^{T}$ and so we have that $\rho$ and $u$ switch roles. In particular, the new transition probability from $v_i$ to $v_j$ is (in terms of the quantities defined in § 4) $\overline {p}_{ij} = {M_{ji}u_j}/{\lambda u_i}$ but the stationary measure on vertices is unchanged.

Proof of Lemma 4.6 Assertion (i) is the well-known Kac lemma [Reference KacKac47, Theorem 2’]. To prove (ii), recall that $\Gamma _{v, w}$ is the set of all paths $\gamma$ from $v$ to $w$ which do not pass through $v$ in their middle. Hence, if we reverse all the paths in this set, we obtain $\overline \Gamma _{v,w}$, the set of all paths $\overline \gamma$ from $w$ to $v$ which do not pass through $v$ in their middle. Note that since almost every path starting at $w$ passes through $v$,

\begin{align*} 1&= \sum_{\overline \gamma \in \overline \Gamma_{v,w}} \overline \mu(\overline \gamma)\\ &= \frac{u_v}{u_w} \sum_{\overline \gamma \in \overline \Gamma_{v,w}} \lambda^{-|\overline \gamma|}, \end{align*}

where $\overline \mu (\overline \gamma )$ is the product of the measures of the edges of $\overline \gamma$ with respect to $\overline p$ and we have used our previous observation about $\overline p$.

Using this and the fact that

\begin{align*} \sum_{\overline \gamma \in \overline \Gamma_{v,w}} \lambda^{-|\overline \gamma|} = \sum_{ \gamma \in \Gamma_{v,w}} \lambda^{-| \gamma|}, \end{align*}

we compute

\begin{align*} n_w &= \nu_w(\mathcal{M}) \\ &= \sum_{\gamma \in \Gamma_{v, w}} \mu(\gamma) = \frac{\rho_w}{\rho_v} \sum_{\gamma \in \Gamma_{v, w}} \lambda^{-|\gamma|}\\ &= \frac{\rho_w}{\rho_v} \cdot \frac{u_w}{u_v}= \frac{\pi_w}{\pi_v}. \end{align*}

Hence, the lemma follows from (i).

4.5 The central limit theorem for the Markov chain

We are now in a position to prove Theorem 4.1. By Melbourne and Török [Reference Melbourne and TörökMT04, Theorem 1.1], we have the following proposition.

Proposition 4.7 Let $\phi \colon \Omega \times \overline {X}^{h} \to \mathbb {R}$ belong to $L^{b}(\Omega \times \overline {X}^{h}, \overline {\nu })$ for some $b > 2$, and let $m := \int \phi \, d\overline {\nu }$. Define $\Phi \colon \Omega _v \times \overline X^{h} \to \mathbb {R}$ as $\Phi (\omega, \xi ) := \sum _{k = 0}^{r(\omega ) -1} \phi (T^{k}(\omega,\xi )) - m r(\omega )$, and suppose that

\begin{align*} \frac{\sum_{j = 0}^{n-1} \Phi \circ T_v^{j} }{\sqrt{n}} \end{align*}

converges to a normal distribution in probability on $(\Omega _v \times \overline {X}^{h}, \mu _v^{\mathbb {N}} \otimes \nu _v)$. Then the sequence

\begin{align*} \frac{\sum_{j = 0}^{n-1} \phi \circ T^{j} - n m}{\sqrt{n}} \end{align*}

converges to a normal distribution in probability on $(\Omega \times \overline {X}^{h}, \overline {\nu })$.

Proposition 4.8 There exist $\ell, \sigma$ such that for any continuous, compactly supported $F \colon \mathbb {R} \to \mathbb {R}$ one has

\begin{align*} \int_{\Omega \times \overline{X}^{h}} F\bigg(\frac{\beta_\xi(o, g_1\cdots g_n o) - \ell n}{\sqrt{n}}\bigg) d \overline{\nu}(\omega, \xi) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t), \end{align*}

as $n \to \infty$.

Proof. Let us apply the previous proposition with $\phi = f$ where $f \colon \Omega \times \overline X^{h} \to \mathbb {R}$ is defined as $f(\omega, \xi ) := \beta _\xi (o, g_1 o)$. Then by definition of $\Phi$ and $f$, Lemma 3.6 gives that for every $\omega \in \Omega _v$,

\begin{align*} \Phi(\omega, \xi) &= \sum_{k = 0}^{r(\omega) - 1} f(T^{k}(\omega,\xi)) - \ell r(\omega)\\ &= \beta_{\xi}(o, w_{r(\omega)} o) - \ell r(\omega) =: f_v(\omega, \xi), \end{align*}

where

\begin{align*} \ell = m = \frac{\int \beta_\xi(o, go) \, d\mu_v(g)\, d\nu_v(\xi)}{\int \Vert g \Vert \, d\mu_v(g)}. \end{align*}

Now, by Corollary 3.3, integrating in $d\nu _v$, we have for some $\sigma _1 \geq 0$,

\begin{align*} \int_{G \times \overline{X}^{h}} F\bigg(\frac{\beta_\xi(o, go) - \ell \Vert g \Vert}{\sqrt{n}}\bigg) d\mu_v^{*n}(g)\, d\nu_v(\xi) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_{\sigma_1}(t). \end{align*}

Note, moreover, that $\beta _\xi (o, l_1 \cdots l_n o) - \ell \Vert l_1 \cdots l_n \Vert = \sum _{j = 0}^{n-1} f_v (T_v^{j}(\omega, \xi ))$, hence we can rewrite the above equation as

\begin{align*} \int_{\Omega_v \times \overline{X}^{h}} F\bigg(\frac{\sum_{j = 0}^{n-1} f_v( T_v^{j}(\omega, \xi))}{\sqrt{n}}\bigg) d(\mu_v^{\mathbb{N}} \otimes \nu_v)(\omega, \xi) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_{\sigma_1}(t). \end{align*}

Thus, by Proposition 4.7 and the above calculation, we also have (for some different $\sigma$)

\begin{align*} \int_{\Omega \times \overline{X}^{h}} F\bigg(\frac{\sum_{j = 0}^{n-1} f \circ T^{j}(\omega, \xi) - \ell n}{\sqrt{n}}\bigg) d \overline{\nu}(\omega, \xi) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t). \end{align*}

The claim follows by again using that, by Lemma 3.6, we have $\sum _{j = 0}^{n-1} f \circ T^{j}(\omega, \xi ) = \beta _\xi (o, g_1 \cdots g_n o)$.

We will now need to go from the CLT for the Busemann cocycle to that for displacement. To do so, we use the following variation of [Reference Benoist and QuintBQ16, Proposition 3.3].

By [Reference Maher and TiozzoMT18, Proposition 4.4] and the fact that $\Gamma _v^{-1}$ is nonelementary (Lemma 2.3), we have $\nu _v(\overline {X}^{h}_\infty ) = 1$ for any vertex $v$.

Lemma 4.9 For any $\epsilon > 0$ there exists $T$ such that for all vertices $w$ in $\Gamma$, all $\xi \in \overline {X}^{h}_\infty$ and all $n \geq 1$ we have

\begin{align*} \mathbb{P}_w (\omega: |d(o, g_1 \cdots g_n o) - \beta_\xi(o, g_1 \cdots g_n o) | \leq T) \geq 1 - \epsilon. \end{align*}

Proof. Recall that by [Reference Maher and TiozzoMT18, § 3.3] there exists a $G$-equivariant map $\pi \colon \overline {X}^{h}_\infty \to \partial X$, where $\partial X$ is the Gromov boundary. Then, by definition of Gromov product and $\delta$-hyperbolicity, we have

(3)\begin{equation} d(o, go) - \beta_{\xi}(o, go) = 2 (go, \pi(\xi))_o + O(\delta) \end{equation}

for any $\xi \in \overline {X}^{h}_\infty$ (see, for example, [Reference HorbezHor18, Lemma 2.4]). Now, since the pushforward of the stationary measure $\mathbb {P}_w$ for the Markov chain starting at $w$ to the Gromov boundary of $X$ is not atomic [Reference Gekhtman, Taylor and TiozzoGTT20, Lemma 4.2], we have that for every $\epsilon > 0$ there exists $T$ such that

\begin{align*} \mathbb{P}_w\Bigl({\omega \in \Omega_w: \sup_{n \geq 1} (w_n o, \pi(\xi))_o \leq T}\Bigr) \geq 1 - \epsilon \end{align*}

for all $\xi \in \overline {X}^{h}_\infty$ and for all $w$. This, combined with (3), yields the desired estimate.

Proof of Theorem 4.1 Let $F \colon \mathbb {R} \to \mathbb {R}$ be continuous with compact support. Since $F$ is uniformly continuous and by Lemma 4.9, for any $\eta > 0$ there exists $n_0$ such that for any $n \geq n_0$, any $w$ and any $\xi \in \overline {X}^{h}_\infty$ one has

\begin{align*} \bigg|F\bigg(\frac{d(o, w_n o) - \ell n}{\sqrt{n}}\bigg) - F\bigg(\frac{\beta_\xi(o, w_n o) - \ell n}{\sqrt{n}}\bigg)\bigg| < \eta \end{align*}

with probability $\mathbb {P}_w$ at least $1 - \epsilon$. Thus, since $\overline {\nu } = ({1}/{R}) \sum _w \mathbb {P}_w \otimes \nu _w$, for any $\eta > 0$ there exists $n_0$ such that for any $n \geq n_0$ we have

(4)\begin{equation} \bigg| F\bigg(\frac{d(o, g_1 \cdots g_n o) - \ell n}{\sqrt{n}}\bigg) - F\bigg(\frac{\beta_\xi(o, g_1 \cdots g_n o) - \ell n}{\sqrt{n}}\bigg)\bigg| < \eta \end{equation}

on a subset of $\Omega \times \overline {X}^{h}$ of $\overline {\nu }$-measure $\geq 1 - \epsilon$. On the other hand, by Proposition 4.8, we have

\begin{align*} \int_{\Omega \times \overline{X}^{h}} F\bigg(\frac{\beta_\xi(o, g_1\cdots g_n o) - \ell n}{\sqrt{n}}\bigg) d \overline{\nu}(\omega, \xi) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t). \end{align*}

Hence,

\begin{align*} \int_{\Omega \times \overline{X}^{h}} F\bigg(\frac{d(o, g_1\cdots g_n o) - \ell n}{\sqrt{n}}\bigg) d \overline{\nu}(\omega, \xi) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t). \end{align*}

Since the integrand does not depend on $\xi$, then we also have

\begin{align*} \int_\Omega F\bigg(\frac{d(o, g_1\cdots g_n o) - \ell n}{\sqrt{n}}\bigg) d P(\omega) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t), \end{align*}

where $P$ is the pushforward of $\overline {\nu }$ to $\Omega$. Finally, since $\overline {\nu } = ({1}/{R}) \sum _w \mathbb {P}_w \otimes \nu _w$, the pushforward of $\overline {\nu }$ to $\Omega$ equals $P = \sum _w \frac {n_w}{R} \mathbb {P}_w$, where $n_w = \nu _w(M)$. Hence, Lemma 4.6 implies that $P = \sum _w \pi _w \mathbb {P}_w = \mathbb {P}$, thus we also have

\begin{align*} \int_\Omega F\bigg(\frac{d(o, g_1 \cdots g_n o) - \ell n}{\sqrt{n}}\bigg) d\mathbb{P}(\omega) \to \int_{\mathbb{R}} F(t) \, d \mathcal{N}_\sigma(t) \end{align*}

as required.

5.1 Uniformly bicontinuous functions

Let us begin by introducing a class of functions that are well behaved under bounded perturbations in the group.

Definition 5.2 A function $f \colon \Omega ^{*} \to \mathbb {R}$ is uniformly bicontinuous if, for any finite set $B \subseteq G$ and any $\eta > 0$, there exists $N\ge 0$ such that if $\Vert g \Vert \geq N$ and $b_1 \overline g b_2 = \overline h$ in $G$ for some $b_1, b_2 \in B$, then

\begin{align*} | f(g) - f (h)| < \eta. \end{align*}

In our application, the uniformly bicontinuous property will be a consequence of the fact that displacement is Lipschitz in both the right and left word metric on $G$. This definition is inspired by [Reference Calegari and FujiwaraCF10] (in particular, the proof of [Reference Calegari and FujiwaraCF10, Lemma 4.24]), where the bi-Lipschitz property is extensively used.

We next introduce the primary functions of interest used throughout this section. Define the following functions on $\Omega ^{*}$: for any $\ell \in \mathbb {R}$,

\begin{align*} \varphi(g) := \frac{d(o, g o) - \ell \Vert g \Vert}{\sqrt{\Vert g \Vert}},\quad \psi(g) := \frac{d(o, g o)}{\Vert g \Vert}. \end{align*}

We remark that the constant $\ell$ will be chosen once and for all after the proof of Lemma 5.6.

Proof. Fix a finite set $B \subset G$, and let $\mathcal {B}$ be the resulting constant from Definition 2.2. Suppose that $\overline {h} = b_1 \overline {g} b_2$ for some $b_1, b_2 \in B$. Then $|\Vert g \Vert - \Vert h \Vert | \leq \mathcal {B}$ and by the triangle inequality $|d(o, g o) - d(o, h o)| \leq \mathcal {B}_1$ where $\mathcal {B}_1 := 2 \max _{b \in B} d(o, bo)$. Finally, denote by $\mathcal {L}$ the Lipschitz constant so that $d(o, g o) \leq \mathcal {L} \Vert g \Vert$ for any $g \in \Omega ^{*}$.

(1) By the above estimates,

\begin{align*} \bigg|\frac{d(o, h o)}{\Vert h \Vert} - \frac{d(o, g o)}{\Vert g \Vert}\bigg| &= \frac{| (d(o, ho) - d(o, go)) \Vert g \Vert + d(o,go) (\Vert g \Vert - \Vert h \Vert )|}{\Vert g \Vert \Vert h \Vert} \\ &\leq \frac{| d(o, ho) - d(o, go)| }{\Vert h \Vert} + \frac{ d(o,go) }{\Vert g \Vert } \cdot \frac{| \Vert g \Vert - \Vert h \Vert |}{ \Vert h \Vert} \\ &\leq \frac{\mathcal{B}_1 + \mathcal{L} \mathcal{B}}{\Vert g \Vert - \mathcal{B}}, \end{align*}

and the right-hand side tends to $0$ as $\Vert g \Vert \to \infty$.

(2) We can write

\begin{align*} | \varphi(g) - \varphi(h)| = \frac{x}{\sqrt{n}} - \frac{x + y}{\sqrt{n + d}} \end{align*}

where $x = d(o, g o) - \ell \Vert g \Vert$, $y = d(o, h o) - \ell \Vert h \Vert - d(o, g o) + \ell \Vert g \Vert$, $n = \Vert g \Vert$, and $d = \Vert h \Vert - \Vert g \Vert$.

Recall that by the above inequalities $|d| \leq \mathcal {B}$, hence also $|y| \leq \mathcal {B}_1 + \ell \mathcal {B}$ and $|x| \leq (\mathcal {L}+\ell ) \Vert g \Vert$. Thus,

\begin{align*} \bigg|\frac{x}{\sqrt{n}} - \frac{x + y}{\sqrt{n + d}}\bigg| &= \bigg|\frac{x(\sqrt{n+d}-\sqrt{n})}{\sqrt{n (n+d)}} - \frac{y}{\sqrt{n+d}}\bigg| \\ &\leq \frac{|x|}{n} \frac{n}{\sqrt{n(n+d)}} |\sqrt{n+d}-\sqrt{n}| + \frac{|y|}{\sqrt{n+d}} \\ &\leq (\mathcal{L}+\ell) \frac{n}{\sqrt{n(n- \mathcal{B})}} (\sqrt{n+ \mathcal{B}}-\sqrt{n}) + \frac{\mathcal{B}_1 + \ell \mathcal{B}}{\sqrt{n-\mathcal{B}}} \end{align*}

and the right-hand side tends to $0$ uniformly in $n$.

Remark 5.4 (Logarithmic perturbations)

As a consequence of the proof that $\varphi$ is uniformly bicontinuous, we observe that for any $\eta > 0$ there is an $N$ such that if $\Vert g \Vert \geq N$ then for any decomposition $g = g_0 g_1 g_2$ with $\Vert g_0 \Vert, \Vert g_2 \Vert \leq \log N$ we have

\begin{align*} | \varphi(g) - \varphi (g_1)| < \eta. \end{align*}

The main reason why we introduce the bicontinuous functions is the following property. For each $i$, we denote by $\mu _n^{(i)}$ the distribution on the space of paths of length $n$ induced by the Markov measure associated to the maximal component $C_i$, as defined in § 4. In particular, $\mu _n^{(i)}$ is supported on length $n$ paths of $\Gamma$ that are contained in $C_i$.

Lemma 5.5 Suppose that the graph structure $\Gamma$ is thick and coherent. Let $f \colon \Omega ^{*} \to \mathbb {R}$ be a uniformly bicontinuous function, and suppose that for each maximal component $C_i$ there is a finite measure $\mathcal {D}_i$ on $\mathbb {R}$ so that

\begin{align*} f(w_n) \to \mathcal{D}_i \end{align*}

in distribution with respect to the Markov measure on $C_i$. If each $\mathcal {D}_i$ is nonatomic, then there exists $a \ge 1$ such that for any interval $I \subset \mathbb {R}$,

\begin{align*} \frac{1}{a} \mathcal{D}_i(I) \le \mathcal{D}_j(I) \le a \mathcal{D}_i(I). \end{align*}

Similarly, if each $\mathcal {D}_i$ is a Dirac measure, then $\mathcal {D}_i = \mathcal {D}_j$.

Proof. Let $v$ be a vertex of $C_i$, and let $C_j$ be another maximal component. Let $B\subset G$ be the finite set given by thickness of $\Gamma$ (Definition 2.1) and let $\mathcal {B}$ be the resulting constant from Definition 2.2. Similar to the proof of Lemma 2.3, by thickness and coherence we have

\begin{align*} \overline{S_n^{(j)}} \subseteq \bigcup_{b_1, b_2 \in B} \bigcup_{|k| \leq \mathcal{B}} b_1 (\overline{\Gamma_v \cap S_{n + k}}) b_2 \end{align*}

where $S_n^{(j)}$ is the set of paths of length $n$ which lie entirely in $C_j$. Hence, by uniform continuity, for any $\epsilon > 0$ there exists $n_0$ such that for all $n \geq n_0$,

\begin{align*} \#\{ g \in S_n^{(j)}: f(g) \in [x,y] \} \leq \mathfrak{b} |B|^{2} \sum_{|k| \leq \mathcal{B}} \#\{ g \in \Gamma_v \cap S_{n+k}: f(g) \in [x-\epsilon,y+\epsilon] \}, \end{align*}

for any interval $I = [x,y]$. Here, we have used that the evaluation map is at most $\mathfrak {b}$-to-one. Now, note that there exists $C > 0$ such that

(5)\begin{equation} C^{-1} \frac{\#(A \cap S_n^{(i)})}{\lambda^{n}} \leq \mu_n^{(i)}(A) \leq C \frac{\#(A \cap S_n^{(i)})}{\lambda^{n}} \end{equation}

for any $i$, any $n$ and any set $A$. Hence, by noting that $\Gamma _v \cap S_{n+k} \subseteq S_{n+k}^{(i)}$,

\begin{align*} \mu_n^{(j)}( g : f(g) \in [x,y] ) \leq \mathfrak{b}|B|^{2} C^{2} \sum_{|k| \leq \mathcal{B}} \mu_{n+k}^{(i)}( g: f(g) \in [x-\epsilon,y+\epsilon] ). \end{align*}

Now taking limits and using the portmanteau theorem, we obtain that

\begin{align*} \mathcal{D}_j((x,y)) \le 2\mathcal{B}\mathfrak{b}|B|^{2} C^{2} \mathcal{D}_i([x-\epsilon,y+\epsilon]), \end{align*}

for any $\epsilon > 0$. So if $c = 2\mathcal {B}\mathfrak {b}|B|^{2} C^{2}$, we have that $\mathcal {D}_j((x,y)) \le c \mathcal {D}_i([x,y])$ because $\mathcal {D}_i([x,y]) = \bigcap _{\epsilon > 0} \mathcal {D}_i([x-\epsilon,y+\epsilon ])$ for any finite measure.

Now if each $\mathcal {D}_i$ is nonatomic, we get $\mathcal {D}_j([x,y]) \le c \mathcal {D}_i([x,y])$, and reversing the roles of $i$ and $j$ completes the proof in this case. Similarly, if $\mathcal {D}_i$ and $\mathcal {D}_j$ were distinct Dirac measures, we would obtain a contradiction by letting $(x,y)$ be a small interval about the atom for $\mathcal {D}_j$ and taking $\epsilon$ small enough so that $[x-\epsilon,y+\epsilon ]$ does not contain the atom for $\mathcal {D}_i$. This completes the proof.

5.2 Uniqueness of drift

We now show that all maximal components of $\Gamma$ determine the same drift.

Proof. By the subadditive ergodic theorem, for each maximal component $C_i$ we have

\begin{align*} \lim_{n \to \infty} \frac{d(o, w_n o)}{n} = \ell_i \end{align*}

almost surely (and hence in distribution) with respect to the measure $\mu _n^{(i)}$. Since $\psi (g) = {d(o, g o)}/{\Vert g \Vert }$ is uniformly bicontinuous by Lemma 5.3, the claim then follows by Lemma 5.5.

By the above lemma, we now define $\varphi$ using $\ell = \ell _i$ for any (equivalently all) $i$.

5.3 Uniqueness of variance

To show that all maximal components of $\Gamma$ determine the same variance, we use the simple fact that distinct normal distributions with mean $0$ can be distinguished by the decay of their tails.

Proof. We have already shown that for $\ell$ fixed as above and any maximal component $C_i$ of $\Gamma$,

\begin{align*} \phi(w_n) \to \mathcal{N}_{\sigma_i} \end{align*}

in distribution with respect to the Markov measure on $C_i$ (Theorem 4.1). Since $\phi$ is uniformly bicontinuous by Lemma 5.3, Lemma 5.5 states that there exists an $a\ge 1$ so that for any interval $I$ in $\mathbb {R}$,

\begin{align*} \frac{1}{a} \int_I d\mathcal{N}_{\sigma_i} \le \int_I d\mathcal{N}_{\sigma_j} \le a \int_I d\mathcal{N}_{\sigma_i}. \end{align*}

But since this holds for every interval, Lemma 5.8 implies that $\sigma _i = \sigma _j$ as required.

Lemma 5.8 If there exists $a \geq 1$ such that

(6)\begin{equation} \frac{1}{a} \int_I d\mathcal{N}_{\sigma_i} \le \int_I d\mathcal{N}_{\sigma_j} \le a \int_I d\mathcal{N}_{\sigma_i} \end{equation}

for any interval $I$, then $\sigma _i = \sigma _j$.

Proof. If we let

\begin{align*} I_{n, \sigma} := \int_n^{n+1} d\mathcal{N}_{\sigma} = \frac{1}{\sqrt{2 \pi} \sigma} \int_n^{n+1} e^{-({t^{2}}/{2 \sigma^{2}})}\, dt \end{align*}

we get

\begin{align*} \frac{1}{\sqrt{2 \pi} \sigma} e^{-({(n+1)^{2}}/{2 \sigma^{2}})} \leq I_{n, \sigma} \leq \frac{1}{\sqrt{2 \pi} \sigma} e^{-({n^{2}}/{2 \sigma^{2}})} \end{align*}

which yields

(7)\begin{equation} \lim_{n \to \infty} \frac{\log I_{n, \sigma}}{n^{2}} = - \frac{1}{2 \sigma^{2}}. \end{equation}

Then, if (6) holds, then

\begin{align*} \frac{1}{a} \leq \frac{I_{n, \sigma_i}}{I_{n, \sigma_j}} \leq a \end{align*}

hence, by (7)

\begin{align*} - \frac{1}{2 \sigma_i^{2}} = \lim_{n \to \infty} \frac{\log I_{n, \sigma_i }}{n^{2}} = \lim_{n \to \infty} \frac{\log I_{n, \sigma_j}}{n^{2}} = - \frac{1}{2 \sigma_j^{2}} \end{align*}

which yields $\sigma _i = \sigma _j$.

6.1 Convergence to the Markov measure

So far our work has been for maximal components of a semisimple graph structure. In this section we consider the whole graph structure, still in the semisimple case.

Let $\Gamma$ be a semisimple graph structure for $G$ with transition matrix $M$ of spectral radius $\lambda > 1$. Let $v_i$ be the vertices of the graph, and let $v_0$ be a vertex of large growth, which we take as the initial vertex. Then recall that $e_i^{T} M^{n} e_j$ is the number of paths of length $n$ from $v_i$ to $v_j$. Since $M$ is semisimple, the limit

\begin{align*} M_\infty := \lim_{n \to \infty} \frac{M^{n}}{\lambda^{n}} \end{align*}

exists. In particular, in keeping with notation at the beginning of § 4, we denote by $e_i$ the $i$th vector of the standard basis, and define

\begin{align*} \rho_i := \lim_{n \to \infty} \frac{e_i^{T} M^{n} 1}{\lambda^{n}} \quad \text{and} \quad u_i := \lim_{n \to \infty} \frac{e_0^{T} M^{n} e_i}{\lambda^{n}}. \end{align*}

By construction, $\rho = (\rho _i)$ satisfies $\rho = M_\infty 1$ and $M \rho = \lambda \rho$, while $u = (u_i)$ satisfies $u^{T} M = \lambda u^{T}$. Finally, $\sum _i u_i = \rho _0$ and $\sum _i u_i \rho _i = \rho _0$.

Note that vertices $v_i$ for which $\rho _i > 0$ and $u_i > 0$ are precisely vertices of components of maximal growth. The large growth vertices are those with $\rho _i >0$.

As before, we use a standard construction to define a Markov measure $\mathbb {P}$ on the space $\Omega$ of infinite paths starting at any vertex of $\Gamma$. First define the initial distribution of the Markov chain to start at vertex $v_i$ with probability $\pi _i := {u_i \rho _i}/{\rho _0}$. Then assign to an edge from $v_i$ to $v_j$ the probability $ {\rho _j }/{\lambda \rho _i}$ so that the transition probability from $v_i$ to $v_j$ is $p_{ij} := {M_{ij} \rho _j }/{\lambda \rho _i}$. Obviously, $\mathbb {P}$ is supported on paths that are entirely contained in components of maximal growth. We denote by $\mathbb {P}_n$ the distribution on the space of paths of length $n$ induced by the Markov measure $\mathbb {P}$.

The following result relates the Markov measure on the semisimple graph structure to the counting measure. For its statement, let $v_0$ be any vertex of large growth. For each $n$, consider the path given by selecting uniformly a path $\gamma$ starting at $v_0$ of length $n$, and take its subpath $\tilde {\gamma }$ from position $\lfloor \log n \rfloor$ to position $n - \lfloor \log n \rfloor$. To avoid writing the integer part every time, we set $\lg (n) := \lfloor \log n \rfloor$. Let $\tilde {\lambda }_n$ denote the distribution of $\tilde {\gamma }$.

Lemma 6.2 With notation as above, the total variation

\begin{align*} \Vert \mathbb{P}_{n-2\lg(n)} - \tilde{\lambda}_{n}\Vert_{TV} \to 0 \end{align*}

as $n \to \infty$.

Proof. Denote $n' := n - 2 \lg (n)$. Let $\gamma$ be a path in the graph, starting at $v_i$ and ending at $v_j$. Then by definition the proportion of paths of length $n$, starting at $v_0$, that have $\gamma$ as ‘middle subpath’ of length $n'$ is

\begin{align*} \tilde{\lambda}_n(\gamma) = \frac{(e_0^{T} M^{\lg(n)} e_i) (e_j^{T} M^{\lg(n)} 1)}{e_0^{T} M^{n} 1}. \end{align*}

On the other hand,

\[ \mathbb{P}_{n'}(\gamma) = \left\{\begin{array}{@{}ll} \dfrac{\pi_i \rho_j}{\rho_i \lambda^{n'}} & \text{if }v_i\text{ has large growth,} \\ 0 & \text{otherwise,} \end{array}\right. \]

which is nonzero if both $v_i$ and $v_j$ belong to a maximal component. In this case,

\begin{align*} \frac{d\mathbb{P}_{n'}}{d \tilde{\lambda}_{n}}(\gamma) &= \frac{\lambda^{\lg(n)}}{e_0^{T} M^{\lg(n)}e_i}\cdot \frac{\lambda^{\lg(n)}}{e_j^{T} M^{\lg(n)} 1} \cdot \frac{e_0^{T} M^{n} 1}{\lambda^{n}} \cdot \frac{\pi_i \rho_j}{\rho_i}\\ &\longrightarrow \frac{1}{u_i} \cdot \frac{1}{\rho_j} \cdot \frac{\rho_0}{1} \cdot \frac{\pi_i \rho_j}{\rho_i} =1 \end{align*}

using that $\pi _i = {u_i \rho _i}/{\rho _0}$. Moreover, if $S_n^{i,j}$ denotes the set of paths of length $n'$ from $v_i$ to $v_j$, we have

\begin{align*} \tilde{\lambda}_{n}(S_n^{i,j}) &= \frac{(e_i^{T} M^{n'} e_j) (e_0^{T} M^{\lg(n)} e_i) (e_j^{T} M^{\lg(n)} 1)}{e_0^{T} M^{n} 1} \\ &\leq \frac{(e_i^{T} M^{n'} 1) (e_0^{T} M^{\lg(n)} e_i) (e_j^{T} M^{\lg(n)} 1)}{e_0^{T} M^{n} 1} \\ &\to \frac{\rho_i u_i \rho_j}{\rho_0}, \end{align*}

hence such a probability tends to $0$ unless both $v_i$ and $v_j$ belong to a maximal component.

Finally, if we denote by $\mathcal {L}_n$ the set of paths of length $n'$ which lie entirely in a maximal component, we have for any set $A$,

\begin{align*} | \mathbb{P}_{n'}(A) - \tilde{\lambda}_{n}(A)| &\leq \sum_{x \in A \cap \mathcal{L}_n} \bigg|\frac{\mathbb{P}_{n'}(x)}{\tilde{\lambda}_{n}(x)}\tilde{\lambda}_{n}(x) - \tilde{\lambda}_{n}(x)\bigg| + \tilde{\lambda}_{n}(A \setminus \mathcal{L}_n) \\ &\leq \sup_{x \in \mathcal{L}_n} \bigg|\frac{\mathbb{P}_{n'}(x)}{\tilde{\lambda}_{n}(x)} - 1\bigg| + \tilde{\lambda}_{n}(\mathcal{L}_n^{c}) \end{align*}

and both terms tend to $0$ as $n \to \infty$, independently of $A$.

6.2 Central limit theorem for the counting measure in the semisimple case

We are now ready to prove the following result. For its statement, let $S_n$ denote the set of length $n$ paths beginning at the initial vertex $v_0$.

Theorem 6.3 Let $\Gamma$ be a semisimple, thick, coherent graph structure for a nonelementary group $G$ of isometries of a $\delta$-hyperbolic space $(X, d)$, and let $o \in X$ be a base point. Then there exist $\ell \geq 0$, $\sigma \geq 0$ such that for any $a < b$ we have

\begin{align*} \lim_{n \to \infty} \frac{1}{\# S_n} \# \bigg\{g \in S_n: \frac{d(o, g o) - \ell n}{\sqrt{n}} \in [a, b] \bigg\} = \int_a^{b} d \mathcal{N}_\sigma(t). \end{align*}

In the following proof and later on, we will use the notation $N_\sigma (x) := \int _{-\infty }^{x} d\mathcal {N}_\sigma (t)$.

Proof. Let $C_1, \ldots, C_k$ be the maximal components, and let $\mu _n^{(i)}$ be the distribution on the space of paths of length $n$ induced by the Markov measure associated to that component, as in § 4. Theorem 4.1 shows a CLT for all such measures, and by Lemmas 5.6 and 5.7 all such measures have the same drift and variance, which we denote by $\ell$ and $\sigma$.

Now, since the starting probability $(\pi _i)$ in the above construction is nonzero precisely on the set of vertices which belong to a maximal component, there exist weights $c_i \geq 0$ with $\sum _i c_i = 1$ such that

\begin{align*} \mathbb{P}_n = \sum_{i = 1}^{k} c_i \mu_n^{(i)} \end{align*}

for any $n$. Thus, for any $x \in \mathbb {R}$,

(8)\begin{equation} \mathbb{P}_n (g: \varphi(g) \leq x ) = \sum_{i = 1}^{k} c_i \mu_n^{(i)} (g: \varphi(g) \leq x ) \to N_\sigma(x), \end{equation}

where we recall that

\begin{align*} \varphi(g) = \frac{d(o, go) - \ell \Vert g \Vert}{\sqrt{\Vert g \Vert}}. \end{align*}

We now use that the counting measure can be approximated by the distribution on finite paths for the Markov chain. If $g$ is a path of length $n$, we denote $g = g_0 g_1 g_2$ where $g_0$ is the prefix of length $\lg (n)$, $g_1$ is the middle part of length $n - 2 \lg (n)$ and $g_2$ is the final part of length $\lg (n)$. By Remark 5.4, there exists $n_0$ such that

(9)\begin{equation} | \varphi(g) - \varphi(g_1) | \leq \epsilon \end{equation}

for any $n \geq n_0$ and $g$ with $\Vert g \Vert = n$.

Fix $x \in \mathbb {R}$ and $\epsilon > 0$. Then we have

\begin{align*} \lambda_n (g: \varphi(g) \leq x ) &= \lambda_n ( g = g_0 g_1 g_2: \varphi(g) \leq x) \\ &\le \lambda_n ( g = g_0 g_1 g_2: \varphi(g_1) \leq x + \epsilon ) \quad\ \ \, \text{by 9, for $n$ large }\\ &= \tilde{\lambda}_n ( g_1: \varphi(g_1) \leq x + \epsilon ) \quad\quad\quad\quad\quad\ \, \text{by definition of $\tilde{\lambda}_n$} \\ &\le \mathbb{P}_{n - 2 \lg(n)} ( g_1: \varphi(g_1) \leq x + \epsilon ) + \epsilon \quad \text{by Lemma 6.2, for $n$ large.} \end{align*}

Hence, by (8) we obtain

\begin{align*} \limsup_{n \to \infty} \lambda_n ( g: \varphi(g) \leq x) \leq N_\sigma(x + \epsilon) + \epsilon \end{align*}

and, by taking $\epsilon$ smaller and smaller and using the continuity of $N_\sigma$,

\begin{align*} \limsup_{n \to \infty} \lambda_n ( g: \varphi(g) \leq x) \leq N_\sigma(x). \end{align*}

The lower bound follows analogously.

Indeed, the same proof shows the following stronger statement. Let $\lambda _n^{(l)}$ denote the counting measure on the set of paths of length $n$ starting at $v_l$.

Corollary 6.5 Let $\Gamma$ be a semisimple, thick, coherent graph structure for a nonelementary group $G$ of isometries of a $\delta$-hyperbolic space $(X, d)$, and let $o \in X$ be a base point. Then there exist $\ell \geq 0$, $\sigma \geq 0$ such that for any vertex $v_l$ of large growth for $\Gamma$ and any $a < b$ we have

\begin{align*} \lim_{n \to \infty} \lambda_n^{(l)} \bigg(g: \frac{d(o, g o) - \ell \Vert g \Vert}{\sqrt{\Vert g \Vert}} \in [a, b]\bigg) = \int_a^{b} d N_\sigma(t). \end{align*}

Proof. Let us fix a vertex $v_l$ of large growth for $M$. Then we can define a Markov measure $\mathbb {P}^{(l)}$ on the space of infinite paths as follows. The transition probabilities will always be the same $p_{ij} = {\rho _j}/{\lambda ^{p} \rho _i}$, while for each vertex $v_l$ one finds a different set of starting probabilities $\pi _i^{(l)}$ given by

\begin{align*} \pi_i^{(l)} := \frac{u_i^{(l)} \rho_i}{\rho_l}, \quad \text{where}\ u_i^{(l)} := \lim_{n \to \infty} \frac{e_l^{T} M^{n} e_i}{\lambda^{n}}. \end{align*}

Just as before, there exist constants $c_i^{(l)} \geq 0$ such that $\sum _i c_i^{(l)} = 1$ and

\begin{align*} \mathbb{P}_n^{(l)} = \sum_{i =1}^{k} c_i^{(l)} \mu_n^{(i)}. \end{align*}

The proof then proceeds exactly as for Theorem 6.3.

7.1 A central limit theorem for translation length

We now prove a more general version of our second main result, Theorem 1.1(ii).

Theorem 7.4 Let $\Gamma$ be a thick, coherent graph structure for a nonelementary group $G$ of isometries of a $\delta$-hyperbolic space $(X, d)$, let $o \in X$ be a base point, and let $\ell,\sigma$ be as in Theorem 7.3. Then for any $a < b$ we have

\begin{align*} \lim_{n \to \infty} \frac{1}{\# S_n} \# \bigg\{ g \in S_n: \frac{\tau(g) - \ell n}{\sqrt{n}} \in [a, b]\bigg\} = \int_a^{b} d \mathcal{N}_\sigma(t). \end{align*}

Proof. Let us recall that the translation length of an isometry $g$ of a $\delta$-hyperbolic space can be computed by (see, for example, [Reference Maher and TiozzoMT18, Proposition 5.8])

(10)\begin{equation} \tau(g) = d(o, go) - 2(go, g^{-1}o)_o + O(\delta) \end{equation}

where $O(\delta )$ is a constant which only depends on the hyperbolicity constant of $X$. Now, by choosing $f(n) = \epsilon \sqrt {n}$ in [Reference Gekhtman, Taylor and TiozzoGTT20, Proposition 5.8], for any $\epsilon$ we have

\begin{align*} \lambda_n(g: (go, g^{-1} o)_o \leq \epsilon \sqrt{n} ) \to 1 \end{align*}

as $n \to \infty$. The claim then follows by combining this statement and the statement of Theorem 7.3 into formula (10).

7.2 Zero variance

We finally complete our main theorem by characterizing the case where $\sigma = 0$. First, we give a general criterion.

Proposition 7.5 In the hypotheses of Theorem 7.3 we have $\sigma =0$ if and only if there is $C \ge 0$ such that for all finite length paths $g$ in $\Gamma$,

\begin{align*} |d(o, go) - \ell \Vert g \Vert | \le C. \end{align*}

We note that the proposition implies that $\sigma >0$ whenever the action $G \curvearrowright X$ is nonproper.

Proof. Suppose that $\sigma = 0$ for the CLT for the counting measure. Then, by our previous discussion, we have $\sigma = 0$ also for the Markov chain on any maximal components. Then, by Theorem 4.2, we also have $\sigma = 0$ for the random walk on the loop semigroup driven by $\check {\mu }_v$. Hence, as in [Reference Benoist and QuintBQ16, Proof of Theorem 4.7 (b)], for any $n$,

\begin{align*} \frac{1}{n} \int (\eta_0(g, \xi) )^{2} \,d \check{\mu}_v^{*n}(g) \,d \nu_v(\xi) = 0 \end{align*}

where $\eta _0$ is the centering of $\eta$. This implies

\begin{align*} \eta_0(g, \xi) = 0 \end{align*}

for any $g \in \Gamma _v^{-1}$ and $\nu _v$-a.e. $\xi \in \overline X^{h}$. Thus, since $|\eta - \eta _0| \leq 2 \Vert \psi \Vert _\infty$ is bounded, we have

\begin{align*} |\beta_\xi(o, g^{-1} o) - \ell \Vert g \Vert | = |\eta(g, \xi)| \leq 2 \Vert \psi \Vert_\infty, \end{align*}

hence by [Reference HorbezHor18, Corollary 2.3] there exists a constant $C$ for which

\begin{align*} |d(o, go) - \ell \Vert g \Vert | \leq C \end{align*}

for any $g$ in the support of $\check {\mu }_v^{*n}(g)$.

Hence, by thickness we have that for any $g \in \Omega ^{*}$ there exist $b_1, b_2 \in B$ and $h \in \Gamma _v^{-1}$ such that $\overline {h} = b_1 \overline {g} b_2$, thus by coherence and the triangle inequality

\begin{align*} |d(o, go) - \ell \Vert g \Vert | \leq |d(o, ho) - \ell \Vert h \Vert| + \mathcal{B}_1 + \ell \mathcal{B}, \end{align*}

thus there exists a constant $C'$ such that

\begin{align*} |d(o, go) - \ell \Vert g \Vert | \leq C' \end{align*}

for any $g \in \Omega ^{*}$. This completes the proof.

We conclude with a corollary that applies when the graph structure is geodesic. For the action $G \curvearrowright X$, denote the translation length of $h$ by $\tau _X(h)$. We use the notation $\tau _G(h)$ to denote the translation length of $h$ with respect to the word metric $d_G$ induced by the graph structure $\Gamma$:

\begin{align*} \tau_G(h) = \lim_{n\to \infty}\frac{1}{n}d_G(1, h^{n}). \end{align*}

Corollary 7.6 Suppose that $\Gamma$ is a thick geodesic combing of $G$. If $\sigma = 0$ in the CLT, then for all $h \in G$,

\begin{align*} \tau_X(h) = \ell \tau_G(h), \end{align*}

where $\ell$ is the corresponding drift.

Proof. Recall that a geodesic combing is coherent by Lemma 2.4. Let $g_n$ be a path in $\Gamma$ representing $h^{n}$ for $h\in G$. That is, $h^{n} = \overline g_n$. Since the structure is geodesic, $||g_n|| = d_G(1,h^{n})$. Applying Proposition 7.5, we get that

\begin{align*} |d(o, h^{n}o) - \ell d_G(1,h^{n})| = O(1). \end{align*}

The corollary follows after dividing by $n$ and taking a limit.