Hyperfiniteness of boundary actions of cubulated hyperbolic groups

TL;DR

This paper proves that hyperbolic groups acting on CAT(0) cube complexes have boundary actions that are hyperfinite, extending classical results from free groups to a broader class of groups.

Contribution

It establishes the hyperfiniteness of boundary actions for cubulated hyperbolic groups, generalizing previous results from free groups to hyperbolic groups acting on cube complexes.

Findings

Boundary actions of cubulated hyperbolic groups are hyperfinite.

Generalizes Dougherty, Jackson, and Kechris's result from free groups.

Connects geometric group actions with descriptive set theory.

Abstract

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

Equations

$$[\eta ,x)=\{y\in X^{(0)}:\text{y lies on a combinatorial geodesic from x to η}\}$$

$$x$$

$$\eta$$

$$(B,C)\in \Phi ⟺$$

$$\wedge \forall (x,y)\neg xBy\vee \neg yBx$$

$$\wedge \forall (x,y,z)\neg xBy\vee \neg yBz\vee \neg xCz$$

$$\wedge {\forall }_{i=1}^n{x}_i(\underset{i\ne j}{\bigvee }{x}_i=x_j)\vee (\underset{i\ne j}{\bigvee }\neg x_{i}B{x}_j)$$

$$c(\eta ,m,n)=(c({\eta }_m,{\eta }_{m+1}),c({\eta }_{m+1},{\eta }_{m+2}),\dots ,c({\eta }_{m+n-1},{\eta }_{m+n}))\in C^{<\mathbb{N}}$$

$$S^a=\{({\eta }_m,c(\eta ,m,n))\in V\times C^{<\mathbb{N}}:\eta \text{mbox}isacombinatorialgeodesic$$

$$\text{mbox}from{v}_0\text{mbox}toa\text{mbox}andm,n\in \mathbb{N}\}$$

$$T_n^a=\{v\in V:(v,s_n^a)\in S^a\}$$

$$k_n^a=d(v_0,v_n^a)$$

$$H_n(a)=g_n^a{T}_n^a,$$

$$g{g}_n^a{v}_0$$

$$2d(v,gv)$$

$$=d({\gamma }_{m_4},{\gamma }_{m_5})+d(v,{\gamma }_{m_4})+d(v,{\eta }_{m_2})+d({\eta }_{m_2},{\gamma }_{m_5})$$

$$\le 2(d(v,{\gamma }_{m_4})+d(v,{\eta }_{m_2})+d({\eta }_{m_2},{\gamma }_{m_5}))$$

$$\le 2(6\delta ),$$

Full text

The authors would like to acknowledge support from the NCN (Polish National Science Centre) through the grant Harmonia no. 2015/18/M/ST1/00050. Marcin Sabok acknowledges also support from NSERC through the Discovery Grant RGPIN-2015-03738

1. Introduction

The complexity theory for countable Borel equivalence relations has been an active topic of study over the last few decades. By the classical result of Feldman and Moore [FM77], countable Borel equivalence relations correspond to Borel actions of countable groups and there has been a lot of effort to understand how the structure of the actions of a group depends on the group itself.

Recall that if $Z$ is a standard Borel space, then a Borel equivalence relation on $Z$ is an equivalence relation $E\subseteq Z^{2}$ which is Borel in $Z^{2}$. If $E$ and $F$ are Borel equivalence relations on $Z$ and $Y$ respectively, we say that $E$ is Borel-reducible to $F$ (denoted $E\leq_{B}F$) if there is a Borel function $f:Z\to Y$ such that $z_{1}\mathrel{E}z_{2}$ if and only if $f(z_{1})\mathrel{F}f(z_{2})$ for all $z_{1},z_{2}\in Z$ ($f$ is then called a reduction from $E$ to $F$). A smooth equivalence relation is a Borel equivalence relation which is reducible to $\operatorname{id}_{2^{\mathbb{N}}}$, the equality relation on the Cantor set. The relation $E_{0}$ is defined on the Cantor set $2^{\mathbb{N}}$ as follows: $x\mathrel{E_{0}}y$ if there exists $n$ such that $x(m)=y(m)$ for all $m>n$. A finite (resp. countable) equivalence relation is an equivalence relation whose classes are finite (resp. countable). An equivalence relation $E$ on $X$ is hyperfinite (resp. hypersmooth) if there is a sequence $F_{n}$ of finite (resp. smooth) equivalence relations on $X$ such that $F_{n}\subseteq F_{n+1}$ and $E=\bigcup_{n}F_{n}$. Note that if $E\leq_{B}F$ and $F$ is hypersmooth, then $E$ is also hypersmooth.

Among countable equivalence relations, hyperfinite equivalence relations are exactly those which are Borel-reducible to $E_{0}$ [DJK94]. The classical dichotomy of Harrington, Kechris and Louveau [HKL90] implies that if a countable Borel equivalence relation is not smooth, then $E_{0}$ is Borel-reducible to it. Interestingly, a very recent result of Conley and Miller [CM17] implies that among countable Borel equivalence relations which are not hyperfinite there is no countable basis with respect to Borel-reducibility.

Hyperfinite equivalence relations have particular structure, observed by Slaman and Steel and independently by Weiss (see [Gao09, Theorem 7.2.4]). An equivalence relation $E$ on $Z$ is hyperfinite if and only if there exists a Borel action of the group of integers $\mathbb{Z}$ on $Z$ which induces $E$ as its orbit equivalence relation. In recent years, there has been a lot of effort to understand which groups induce hyperfinite equivalence relations. For instance, Gao and Jackson [GJ15] showed that Borel actions of all Abelian groups induce hyperfinite equivalence relations. It is still unknown if all Borel actions of amenable groups induce hyperfinite equivalence relations.

In this paper we are mainly interested in actions of hyperbolic groups. Recall that a geodesic metric space $X$ is hyperbolic if there exists $\delta>0$ such that all geodesic triangles in $X$ are $\delta$-thin, i.e. each of their sides is contained in the $\delta$-neighborhood of the union of the other two sides. In such case we also say that $X$ is $\delta$-hyperbolic. A finitely generated group $G$ is hyperbolic if its Cayley graph (w.r.t. an arbitrary finite generating set) is hyperbolic. An isometric action of a group $G$ on a metric space $X$ is proper if for every compact subset $K\subseteq X$ the set $\{g\in G:gK\cap K\neq\emptyset\}$ is finite. An isometric action of $G$ on $X$ is cocompact if there exists a compact subset $A$ of $X$ such that $GA=X$. If $X$ is a combinatorial complex, then an isometric action of a group on $X$ is proper if and only if the stabilizers of all vertices are finite. Similarly, an action on a combinatorial complex is cocompact if and only if there are finitely many orbits of vertices. An action of a group is called geometric if it is both proper and cocompact. If a group $G$ acts geometrically on a geodesic metric space $X$ by isometries, then $G$ is hyperbolic if and only if $X$ is hyperbolic, since hyperbolicity is invariant under quasi-isometries.

Given a geodesic hyperbolic space $X$ we denote by $\partial X$ its Gromov boundary (for definition see e.g. [KB02]). Any geometric action of a hyperbolic group $G$ on a hyperbolic space $X$ induces a natural action of $G$ on $\partial X$ by homeomorphisms. If $X$ is the Cayley graph of a hyperbolic group $G$, then the Gromov boudary of $X$ is called the Gromov boundary of the group $G$.

Hyperbolic groups often admit geometric actions on CAT$(0)$ cube complexes. Recall that a cube complex is obtained by taking a disjoint collection of unit cubes in Euclidean spaces of various dimensions, and gluing them isometrically along their faces. A geodesic metric space $X$ is CAT$(0)$ if for every geodesic triangle $\Delta$ in $X$ and a comparison triangle $\Delta^{\prime}$ in the Euclidean plane, with sides of the same length as the sides of $\Delta$, the distances between points on $\Delta$ are less than or equal to the distances between the corresponding points on $\Delta^{\prime}$. This is one way of saying that a metric space has nonpositive curvature. For more details on CAT$(0)$ cube complexes see Section 2.

If a hyperbolic group admits a geometric action on a CAT$(0)$ cube complex, then we say that it is cubulated. Examples of cubulated hyperbolic groups include

• •

fundamental groups of hyperbolic surfaces and hyperbolic closed 3-manifolds (Kahn and Markovic [KM12] and Bergeron and Wise [BW12]),

• •

uniform hyperbolic lattices of ”simple type” (Haglund and Wise [HW12]),

• •

hyperbolic Coxeter groups (Niblo and Reeves [NR97] and Caprace and Mühlherr [CM05]),

• •

$C^{\prime}(1/6)$ or $C^{\prime}(1/4)$-$T(4)$ metric small cancellation groups (Wise [Wis04]),

• •

certain cubical small cancellation groups (Wise [Wis17]),

• •

Gromov’s random groups with density $<1/6$ (Ollivier and Wise [OW11]),

• •

hyperbolic free-by-cyclic groups (Hagen and Wise [HW16] in the irreducible case and [HW15] in the general case).

It is worth noting that cubulations of hyperbolic groups played important role in recent breakthroughs on the Virtual Haken Conjecture by Agol [Ago13] and Wise [Wis17]. The main result of this paper is the following.

Theorem 1.1**.**

If a hyperbolic group $G$ acts geometrically on a $\mathrm{CAT}(0)$ cube complex $X$, then the induced action on $\partial X$ is hyperfinite.

Note that if $G$ acts geometrically on $X$ and $Y$, then there is a $G$-equivariant homeomorphism of $\partial X$ and $\partial Y$ [Gro87]. Hence, the above theorem implies the following.

Corollary 1.2**.**

If $G$ is a hyperbolic cubulated group, then its natural boundary action on $\partial G$ is hyperfinite.

The boundary actions of hyperbolic groups have been studied from the perspective of their complexity. Recall that if $\mu$ is a probability measure on a standard Borel space $X$ and $E$ is a countable Borel equivalence relation on $X$, then $E$ is called $\mu$-hyperfinite if there exists a $\mu$-conull set $A\subseteq X$ such that $E\cap A^{2}$ is hyperfinite. A Borel probability measure $\mu$ is $E$-quasi-invariant if it is quasi-invariant with respect to any group action inducing $E$. Kechris and Miller [KM04, Corollary 10.2] showed that if $E$ is $\mu$-hyperfinite for all $E$-quasi-invariant Borel measures, then $E$ is $\mu$-hyperfinite for all Borel measures $\mu$. It is worth noting, however, that for boundary actions of hyperbolic groups usually there is no unique quasi-invariant measure on the boundary.

In the case of the free group, its boundary action induces the equivalence relation which is Borel bi-reducible with the so-called tail equivalence relation on the Cantor set: $x\mathrel{E_{t}}y$ if $\exists n\,\exists m\,\forall k\ x(n+k)=y(m+k)$. It follows from the results of Connes Feldman and Weiss [CFW81, Corollary 13] and Vershik [Ver78] that if $G$ is the free group, then the action of $G$ on its Gromov boundary (which is the Cantor set) is $\mu$-hyperfinite for every Borel quasi-invariant probability measure. Dougherty Jackson and Kechris [DJK94, Corollary 8.2] showed later that the tail equivalence relation is actually hyperfinite. On the other hand, Adams [Ada94] showed that for every hyperbolic group $G$ the action on $\partial G$ is $\mu$-hyperfinite for all Borel quasi-invariant probability measures. We do not know how to generalize Corollary 1.2 to all hyperbolic groups.

Our proof uses an idea of Dougherty, Jackson and Kechris [DJK94] and the main ingredient of the proof is a result that seems to be interesting on its own right. Given an element a hyperbolic group $G$ acting geometrically on a cube complex $C$, $\gamma\in\partial C$ and an element $x\in C$, define the interval $[x,\gamma)$ to be the set of all vertices of the complex which lie on a geodesic ray in the 1-skeleton of $C$ from $x$ to $\gamma$. We would like to emphasize here that we consider here only the 1-skeleton of $C$ and all geodesics we consider are the combinatorial geodesics, i.e. those taken in the 1-skeleton.

Lemma 1.3**.**

If a hyperbolic group $G$ acts geometrically on a $\mathrm{CAT}(0)$ cube complex $C$ and $\gamma\in\partial C$, then for every $x,y\in C$ the sets $[x,\gamma)$ and $[y,\gamma)$ differ by a finite set.

Theorem 1.1 is obtained using Lemma 1.3 and the following result:

Theorem 1.4**.**

Suppose a hyperbolic group $G$ acts freely and cocompactly on a locally finite graph $V$ such that for every $\gamma\in\partial V$ and for every $x,y\in V$ the sets $[x,\gamma)$ and $[y,\gamma)$ differ by a finite set. Then the action of $G$ on $\partial V$ induces a hyperfinite equivalence relation.

The assumption that $G$ acts freely on $V$ means that the action on the set of vertices of $V$ is free.

Given a fixed finite set of generators for a hyperbolic group $G$, the group acts on its Cayley graph. For $g\in G$ and $\gamma\in\partial G$, the set $[g,\gamma)$ is defined as above. The following question seems natural. 111it has been answered recently in the negative by N. Touikan

Question 1.5**.**

Suppose $G$ is a hyperbolic group with a fixed finite generating set and $\gamma\in\partial G$. Is it true that for any two group elements $g,h\in G$ the sets $[g,\gamma)$ and $[h,\gamma)$ differ by a finite set?

Of course, it may turn out that the answer to the above question depends on the choice of the generating set. Or, more generally, one can ask the following question.

Question 1.6**.**

Is it true that for every hyperbolic group $G$ there exists a locally finite graph $V$ such that $G$ acts geometrically (or even freely and cocompactly) on $V$, and $V$ has the property that $[x,\gamma)$ and $[y,\gamma)$ have finite symmetric difference for every $\gamma\in\partial V$ and $x,y\in V$?

We should add here that the class of groups for which we can prove the positive answer to the above question is limited to groups with the Haagerup property. We do not know any examples of groups which have the property stated in the above question and do not have the Haagerup property.

Acknowledgement. We would like to thank Piotr Przytycki for inspiration and many helpful discussions.

2. $\mathrm{CAT}(0)$ cube complexes

Here we give a summary of several basic properties of CAT$(0)$ cube complexes without proof. For more details we refer the reader to [BH11, Chapter II.5] and [Sag12].

Recall that a cube complex is obtained by taking a disjoint collection of unit cubes in Euclidean spaces of various dimensions, and gluing them isometrically along their faces. In particular, every cube complex has a piecewise Euclidean metric.

A cube complex $X$ is uniformly locally finite if there exists $D>0$ such that each vertex is contained in at most $D$ edges. Note that if $X$ admits a cocompact group action, then it is automatically uniformly locally finite.

Now, for each vertex $v$ in a cube complex $X$, draw an $\varepsilon$-sphere $S_{v}$ around $v$. Note that the cubes of $X$ divide $S_{v}$ into simplices (a priori, these simplices may not be embedded in $S_{v}$, since a cube may not be embedded in $X$). Thus $S_{v}$ has the structure of a combinatorial cell complex which is made of various simplices glued along their faces. This complex is called the link of the vertex $v$.

Recall that a simplicial complex $K$ is flag if every complete subgraph of the 1-skeleton of $K$ is actually the 1-skeleton of a simplex in $K$.

Definition 2.1**.**

A $\operatorname{CAT}(0)$ cube complex is a cube complex which is simply connected and such that the link of each its vertex is a flag simplicial complex.

The above is a combinatorial equivalent definition of CAT$(0)$ property for cube complexes (for more details see [BH11, Definition II.1.2]).

Let $X$ be a CAT$(0)$ cube complex with its piecewise Euclidean metric. A subset of $C\subseteq X$ is convex if for any two points $x,y\in C$, any geodesic segment connecting $x$ and $y$ is contained in $C$. A convex subcomplex of $X$ is a subcomplex which is also convex.

Recall that a mid-cube of $C=[0,1]^{n}$ is a subset of form $f_{i}^{-1}(\{\frac{1}{2}\})$, where $f_{i}$ is one of the coordinate functions.

Definition 2.2**.**

A hyperplane $h$ in $X$ is a subset such that

1. (1)

$h$ is connected. 2. (2)

For each cube $C\subseteq X$, $h\cap C$ is either empty or a mid-cube of $C$.

It was proved by Sageev [Sag95] that for each edge $e\in X$, there exists a unique hyperplane which intersects $e$ in one point. This is called the hyperplane dual to the edge $e$. Actually, given an edge $e$, we can always build locally a piece of hyperplane that cuts through $e$. In order to extend this piece to a hyperplane, one needs to make sure that the this piece does not run into itself when one extends it. It is shown in [Sag95] that this can never happen in a CAT$(0)$ cube complex and thus such extensions exist.

Let $X$ be a CAT$(0)$ cube complex, and let $e\subseteq X$ be an edge. Denote the hyperplane dual to $e$ by $h_{e}$. The following facts about hyperplanes are well-known [Sag95, Sag12].

1. (1)

The hyperplane $h_{e}$ is a convex subset of $X$ and $h_{e}$ with the induced cell structure from $X$ is also a CAT$(0)$ cube complex. 2. (2)

$X\setminus h_{e}$ has exactly two connected components, which are called halfspaces.

Two points in $X$ are separated by a hyperplane $h$ if they are different connected components of $X\setminus h$.

We use the following metric on the 0-skeleton $X^{(0)}$ of $X$. Given two vertices in $X^{(0)}$, the $\ell^{1}$-distance between them is defined to be the length of the shortest path joining them in the 1-skeleton $X^{(1)}$. By [HW08, Lemma 13.1], the $\ell^{1}$-distance between any two vertices is equal to number of hyperplanes separating them.

Given two vertices $u,v\in X$, a combinatorial geodesic between them is an edge path in $X^{(1)}$ joining $u$ and $v$ which realizes the $\ell^{1}$-distance between $u$ and $v$. Note that there may be several different combinatorial geodesics joining $u$ and $v$. By [HW08, Lemma 13.1], an edge path $\omega\subseteq X^{(1)}$ is a combinatorial geodesic if and only if for each pair of different edges $e_{1},e_{2}\subseteq\omega$, the hyperplane dual to $e_{1}$ and the hyperplane dual to $e_{2}$ are different.

An edge path $\omega$ crosses a hyperplane $h\subseteq X$ if there exists an edge $e\subseteq\omega$ such that $h$ is the dual to $e$. So, in other words, $\omega$ is a combinatorial geodesic if and only if there does not exist a hyperplane $h\subseteq X$ such that $\omega$ crosses $h$ more than once.

Let $Y\subseteq X$ be a convex subcomplex (with respect to the piecewise Euclidean metric). Then, by [HW08, Proposition 13.7], $Y$ is also convex with respect to the $\ell^{1}$-metric in the following sense: for any vertices $u,v\in Y^{(0)}$, every combinatorial geodesic joining $u$ and $v$ is contained in $Y$.

In the rest of this paper, we will always use the $\ell^{1}$-metric on $X^{(0)}$ and use $d$ to denote this metric.

Let $Y\subseteq X$ be a convex subcomplex. By [HW08, Lemma 13.8], for any vertex $v\in X$, there exists a unique vertex $u\in Y$ such that $d(u,v)=d(v,Y^{(0)})$. Thus, we have a nearest point projection map $\pi_{Y}:X^{(0)}\to Y^{(0)}$.

Lemma 2.3**.**

Let $Y\subseteq X$ be a convex subcomplex and $v\in X$. Let $\omega$ be a combinatorial geodesic from $v$ to a vertex in $Y$ which realizes the $\ell^{1}$ distance between $v$ and $Y^{(0)}$. Then each hyperplane dual to an edge in $\omega$ separates $v$ from $Y$. Conversely, each hyperplane which separates $v$ from $Y$ is dual to an edge in $\omega$.

Proof.

This is a special case of [HW08, Proposition 13.10]. ∎

The following is a consequence of Lemma 2.3 and the fact that the $\ell^{1}$-distance between any two vertices is equal to number of hyperplanes separating them [HW08, Lemma 13.1].

Corollary 2.4**.**

Let $Y\subseteq X$ be a convex subcomplex. For every $v\in X$ the distance $d(v,Y^{(0)})$ is the number of hyperplanes that separate $v$ from $Y$.

Lemma 2.5**.**

Let $Y\subseteq X$ be a convex subcomplex and $\pi_{Y}:X^{(0)}\to Y^{(0)}$ be the nearest point projection. Given two adjacent vertices $u,v\in X$ write $u^{\prime}=\pi_{Y}(u)$ and $v^{\prime}=\pi_{Y}(v)$. Suppose $h$ is the hyperplane separating $u$ and $v$.

1. (1)

If $h\cap Y=\emptyset$, then $u^{\prime}=v^{\prime}$. 2. (2)

If $h\cap Y\neq\emptyset$, then $u^{\prime}$ and $v^{\prime}$ are adjacent vertices in $Y$. Moreover, the hyperplane separating $u^{\prime}$ and $v^{\prime}$ is exactly $h$.

Proof.

Suppose without loss of generality that $d(v,Y^{(0)})\leq d(u,Y^{(0)})$. Let $\omega_{v}$ and $\omega_{u}$ be the combinatorial geodesics which realize the $\ell^{1}$-distance from $v$ to $Y^{(0)}$ and $u$ to $Y^{(0)}$ respectively.

Suppose first that $h\cap Y=\emptyset$. Then $h\cap\omega_{v}=\emptyset$, otherwise we would have $d(v,Y^{(0)})>d(u,Y^{(0)})$. Thus $h$ separates $u$ from $Y$. Moreover, each hyperplane dual to an edge in $\omega_{v}$ separates $u$ from $Y$. By Corollary 2.4, we have $d(v,Y^{(0)})+1\leq d(u,Y^{(0)})$. On the other hand, the concatenation of the edge $\overline{uv}$ with $\omega_{v}$ has length $\leq d(v,Y^{(0)})+1$. Thus this concatenation realizes the $\ell^{1}$-distance from $u$ to $Y^{(0)}$. It follows that $u^{\prime}=v^{\prime}$.

Now suppose $h\cap Y\neq\emptyset$. First, by Lemma 2.3 we get $\omega_{v}\cap h=\omega_{u}\cap h=\emptyset$ because otherwise $h$ would be dual to some edge in $\omega_{v}$ or $\omega_{u}$ and thus separate $u$ or $v$ from $Y$ and hence be disjoint from $Y$. Let $\omega$ be a geodesic joining $v^{\prime}$ and $u^{\prime}$. Note that $\omega$ is contained in $Y$. The path obtained by concatenating $\omega_{v}$, $\omega$ and $\omega_{u}$ must intersect $h$ because $v$ and $u$ lie on different sides of $h$. Thus $h$ must intersect $\omega$ and thus separate $v^{\prime}$ and $u^{\prime}$. To see that $v^{\prime}$ and $u^{\prime}$ are adjacent, it is enough to show that $h$ is the only hyperplane separating $u^{\prime}$ and $v^{\prime}$. Note, however, that if $h^{\prime}$ is a hyperplane separating $u^{\prime}$ from $v^{\prime}$, then $h^{\prime}$ must intersect the path obtained by contatenating $\omega_{v}$, the edge from $v$ to $u$ and $\omega_{u}$. By Lemma 2.3 we get $h^{\prime}\cap\omega_{v}=h^{\prime}\cap\omega_{u}=\emptyset$ as above. Thus, $h^{\prime}$ intersects the edge from $u$ to $v$ and hence $h^{\prime}=h$.

∎

The above lemma implies that we can naturally extend the nearest point projection map $\pi_{Y}:X^{(0)}\to Y^{(0)}$ to $\pi_{Y}:X^{(1)}\to Y^{(1)}$. The next result follows from from Lemma 2.5.

Corollary 2.6**.**

Let $Y\subseteq X$ be a convex subcomplex. Let $\omega\subseteq X$ be a combinatorial geodesic. Then $\pi_{Y}(\omega)$ is also a combinatorial geodesic.

Note that it is possible that $\pi_{Y}(\omega)$ is a single point.

3. The geodesics lemma

Throughout this section, $X$ will be a uniformly locally finite Gromov-hyperbolic CAT$(0)$ cube complex. Let $\partial X$ be the boundary of $X$.

Definition 3.1**.**

Let $x\in X$ be a vertex and let $\eta\in\partial X$. Define the interval:

[TABLE]

Recall that if $X$ is $\delta$-hyperbolic, then for any $x\in X$ and $\eta\in\partial X$, any two combinatorial geodesic ray $\omega_{1}$ and $\omega_{2}$ from $x$ to $\eta$ satisfy $d(\omega_{1}(t),\omega_{2}(t))\leq 2\delta$ for each $t\geq 0$ and $d_{H}(\omega_{1},\omega_{2})\leq 2\delta$. Here $d_{H}(\omega_{1},\omega_{2})$ denotes the Hausdorff distance between $\omega_{1}$ and $\omega_{2}$.

Now we will prove Lemma 1.3. Note that it suffices to prove the case when $x$ and $y$ are adjacent. Thus in the rest of this section, we will assume $x$ and $y$ are two adjacent vertices in $X$.

Lemma 3.2**.**

Let $h$ be the hyperplane separating $x$ and $y$ and let $\overline{y\eta}$ be a combinatorial geodesic ray from $y$ to $\eta$.

1. (1)

If $\overline{y\eta}$ never crosses $h$, then each vertex of $\overline{y\eta}$ in contained in $[\eta,x)$. 2. (2)

If $\overline{y\eta}$ crosses $h$, let $z\in\overline{y\eta}$ be the first vertex after $\overline{y\eta}$ crosses $h$ and let $\overline{z\eta}\subseteq\overline{y\eta}$ be the ray after $z$. Pick a combinatorial geodesic segment $\overline{xz}$. Then $\overline{xz}$ and $\overline{z\eta}$ fit together to form a combinatorial geodesic ray. In particular, each vertex of $\overline{z\eta}$ is contained in $[\eta,x)$.

Proof.

To see (1), let $\overline{xy}$ be the edge joining $x$ and $y$. Then $h$ is the hyperplane dual to $\overline{xy}$. Since $\overline{y\eta}$ never crosses $h$, then each hyperplane which is dual to some edge of $\overline{y\eta}$ is different from $h$. Thus the concatenation of $\overline{xy}$ and $\overline{y\eta}$ is a combinatorial geodesic ray because all hyperplanes dual to its edges are distinct [HW08, Lemma 13.1]. Thus each vertex of $\overline{y\eta}$ in contained in $[\eta,x)$.

Now we prove (2). Since $\overline{y\eta}$ is a combinatorial geodesic ray, it follows [HW08, Lemma 13.1] that $\overline{z\eta}$ does not cross $h$. Suppose the concatenation of $\overline{xz}$ and $\overline{z\eta}$ is not a combinatorial geodesic ray. Since $\overline{xz}$ and $\overline{z\eta}$ are already geodesic, the only possibility is that there exist edges $e_{1}\subseteq\overline{xz}$ and $e_{2}\subseteq\overline{z\eta}$ such that they are dual to the same hyperplane $h^{\prime}$, again by [HW08, Lemma 13.1]. Let $u_{i}$ and $v_{i}$ be endpoints of $e_{i}$ indicated in the picture below.

[TABLE]

Since $\overline{xz}$ is a combinatorial geodesic, it crosses $h^{\prime}$ only once ([HW08, Lemma 13.1]). Thus the segments $\overline{xu_{1}}$ and $\overline{v_{1}z}$ stay in different sides of $h^{\prime}$. In particular, $x$ and $z$ are in different sides of $h^{\prime}$. Since $\overline{y\eta}$ is a combinatorial geodesic ray, it crosses $h^{\prime}$ only once, thus the segment $\overline{yz}\cup\overline{zu_{2}}$ is in one side of $h^{\prime}$. In particular, $y$ and $z$ are in the same side of $h^{\prime}$. Thus we deduce that $x$ and $y$ are separated by $h^{\prime}$. Since $x$ and $y$ are adjacent, there is only one hyperplane separating them, thus $h^{\prime}=h$. This is a contraction since $\overline{z\eta}$ does not cross $h$. ∎

Proof of Lemma 1.3.

We assume $x$ and $y$ are adjacent. Let $h$ be the hyperplane separating them. We argue by contradiction and suppose there exists a sequence $\{z_{i}:i\in\mathbb{N}\}$ in $[\eta,y)\setminus[\eta,x)$ with $z_{i}\neq z_{j}$ for $i\neq j$. Since $X$ is uniformly locally finite, we can assume $d(z_{i},y)\to\infty$ as $i\to\infty$. Let $\omega_{i}$ be a combinatorial geodesic segment from $y$ to $\eta$ such that $z_{i}\in\omega_{i}$. By Lemma 3.2, each $\omega_{i}$ crosses $h$. Let $\overline{yv_{i}}\subseteq\omega_{i}$ be the segment before $\omega_{i}$ crosses $h$, and let $\overline{u_{i}\eta}\subseteq\omega_{i}$ be the segment after $\omega_{i}$ crosses $h$ (see the picture below). It follows from Lemma 3.2 (2) that $z_{i}\subseteq\overline{yv_{i}}$. In particular, $d(v_{i},y)\to\infty$ as $i\to\infty$.

[TABLE]

Recall that $h$ gives rise to two combinatorial hyperplanes, one containing $x$, which we denote by $h_{x}$, and one containing $y$, which we denote by $h_{y}$. Note that $v_{i}\in h_{y}$ by construction. Since $h_{y}$ is a convex subcomplex, it follows [HW08, Proposition 13.7] that $\overline{yv_{i}}\subseteq h_{y}$. Since $X$ is uniformly locally finite (hence locally compact) and $d(v_{i},y)\to\infty$, up to passing to a subsequence, we can assume the sequence of segments $\{\overline{yv_{i}}\}_{i=1}^{\infty}$ converges to a combinatorial geodesic ray $\omega$. Since $\overline{yv_{i}}\subseteq h_{y}$ for each $i$, $\omega\subseteq h_{y}$. Moreover, by $\delta$-hyperbolicity, the Hausdorff distance between $\omega$ and any of $\omega_{i}$ is less than $2\delta$. Thus $\omega$ is a combinatorial geodesic ray joining $y$ and $\eta$. Since $\omega$ is contained in $h_{y}$ we get that for every $i$ and every vertex $w\in\omega_{i}$ we have $d(w,h_{y})\leq 2\delta$.

Let $\pi:X^{(1)}\to h^{(1)}_{y}$ be the nearest point projection from $X^{(1)}$ to the 1-skeleton of convex subcomplex $h_{y}$. Then $\pi(\omega_{i})$ is a combinatorial geodesic by Corollary 2.6. It follows from the above remarks and the definition of $\pi$ that $d_{H}(\omega_{i},\pi(\omega_{i}))\leq 2\delta$

Thus $\pi(\omega_{i})$ is a combinatorial geodesic ray joining $y$ and $\eta$. Since $\overline{yv_{i}}\subseteq h_{y}$, $\pi(\overline{yv_{i}})=\overline{yv_{i}}$. Thus $\overline{yv_{i}}$ is contained in $\pi(\omega_{i})$. In particular, $z_{i}\in\pi(\omega_{i})$. Since $\pi(\omega_{i})\subseteq h_{y}$, it never crosses $h$, thus Lemma 3.2 (1) implies $z_{i}\in[\eta,x)$, which is a contradiction. ∎

4. Finite Borel equivalence relations

We will use the following standard application of the second reflection theorem [Kec95, Theorem 35.16]. Below, if $E$ is an equivalence relation of $Z$ and $A\subseteq Z$, then $E|A$ denotes $E\cap A\times A$.

Lemma 4.1**.**

Let $Z$ be a Polish space, $A\subseteq Z$ be analytic and let $E$ be an analytic equivalence relation on $Z$ such that there is some $n>1$ such that every $E|A$-class has size less than $n$. Then there is a Borel equivalence relation $F$ on $Z$ with $E|A\subseteq F$ such that every $F$-class has size less than $n$.

Proof.

Note that $G=E|A\cup\{(z,z):z\in Z\}$ is an analytic equivalence relation on $Z$ whose classes have size less than $n$. Now consider $\Phi\subseteq\operatorname{Pow}(Z^{2})\times\operatorname{Pow}(Z^{2})$ defined as follows:

[TABLE]

Note that $\Phi(B,B^{c})$ holds iff $B$ is an equivalence relation on $Z$ whose classes have size less than $n$, so in particular we have $\Phi(G,G^{c})$. Now $\Phi$ is $\mathbf{\Pi}^{1}_{1}$ on $\mathbf{\Pi}^{1}_{1}$, hereditary and continuous upward in the second variable, so by the second reflection theorem [Kec95, Theorem 35.16], there is a Borel set $F\supset G$ such that $\Phi(F,F^{c})$ holds, and we are done. ∎

5. Proof of main theorem

The following fact lets us reduce our problem to the case of free actions.

Lemma 5.1**.**

Every cubulated hyperbolic group has a finite index subgroup acting freely and cocompactly on a $\operatorname{CAT}(0)$-cube complex.

Proof.

If $G$ is a hyperbolic group acting properly and cocompactly on a $\operatorname{CAT}(0)$ cube complex $X$, then by Agol’s theorem [Ago13, Theorem 1.1] (see also Wise [Wis17]) there is a finite index subgroup $F$ acting faithfully and specially on $X$ (see Haglund and Wise [HW08, Definition 3.4] for the definition of special action). Now $F$ embeds into a right-angled Artin group which is torsion-free, so $F$ is torsion-free. Since every stabilizer is finite by properness of the action, it must be trivial since $F$ is torsion-free, and thus $F$ acts freely on $X$. ∎

Proof of Theorem 1.4.

Let $V$ be the set of vertices of the graph. Note that $V$ as a metric space is hyperbolic since the action of $G$ is geometric. Below, by $\partial V$ we denote the Gromov boundary of $V$. Fix $v_{0}\in V$ and fix a total order on $V$ such that $d(v_{0},v)\leq d(v_{0},w)\implies v\leq w$, where $d$ denotes the graph distance on $V$. Fix a transversal $\tilde{V}$ of the action of $G$ on $V$ (the transversal is finite since the action is cocompact). For $v\in V$, we denote by $\tilde{v}$ the unique element of $\tilde{V}$ in the orbit of $v$. By a directed edge of $V$ we mean a pair $(v,v^{\prime})\in V^{2}$ such that there is an edge from $v$ to $v^{\prime}$. We colour the directed edges of $V$ as follows. We assign a distinct colour to every directed edge $(v,v^{\prime})$ with $v\in\tilde{V}$, and this extends uniquely (by freeness) to a $G$-invariant colouring on all directed edges. Let $C$ be the set of colours (which is finite since $V$ is locally finite), and let $c(v,v^{\prime})$ be the colour of $(v,v^{\prime})$. Fix any total order on $C$. This induces a lexicographical order on $C^{<\mathbb{N}}$ (the set of all finite sequences of elements of $C$).

For any combinatorial geodesic $\eta\in V^{<\mathbb{N}}$ and $m,n\in\mathbb{N}$, define:

[TABLE]

For every $a\in\partial V$, define $S^{a}\subseteq V\times C^{<\mathbb{N}}$ as follows:

[TABLE]

Let $s^{a}_{n}\in C^{<\mathbb{N}}$ be the least string of length $n$ which appears infinitely often in $S^{a}$, ie. such that there are infinitely many $v\in V$ for which $(v,s^{a}_{n})\in S^{a}$. Note that each $s^{a}_{n}$ is an initial segment of $s^{a}_{n+1}$. Let

[TABLE]

and let $v^{a}_{n}=\min T^{a}_{n}$ (with respect to the ordering on $V$). Note that every vertex in $T^{a}_{n}$ has an edge coloured by $s^{a}_{1}$ leaving it, so every vertex of $T^{a}_{n}$ is in the same orbit. Let

[TABLE]

and note that $k^{a}_{n}$ is nondecreasing in $n$.

Now let $Z=\{a\in\partial V:k^{a}_{n}\not\to\infty\}$. Then for each $a\in\partial V$, since $k^{a}_{n}\not\to\infty$ and $V$ is discrete, there is a finite set containing all $v^{a}_{n}$, so there is some $v\in V$ which is in $T^{a}_{n}$ for infinitely many $n$. Thus the geodesic class determined by the combinatorial geodesic starting at $\tilde{v}$ (which is determined by $k^{a}_{1}$) and following the colours of $\lim_{n}s^{a}_{n}\in C^{\mathbb{N}}$ is a Borel selector. Thus $E$ is smooth on the saturation $[Z]_{E}$.

Now let $Y=partialV\setminus[Z]_{E}=\{a\in\partial X:\forall bEa\ k^{b}_{n}\to\infty\}$. We will show that $E$ is hyperfinite on $Y$. For each $n\in\mathbb{N}$, and define $H_{n}:\partial V\to 2^{V}$ by

[TABLE]

where $g^{a}_{n}\in G$ is the unique element with $g^{a}_{n}v^{a}_{n}\in\tilde{V}$. Let $E_{n}$ be the equivalence relation on $\operatorname{im}H_{n}$ which is the restriction of the shift action of $G$ on $2^{V}$. We have the following lemma:

Lemma 5.2**.**

There exists $K\in\mathbb{N}$ such that on $\operatorname{im}H_{n}$ the relation $E_{n}$ has equivalence classes of size at most $K$.

Proof.

Let $a,b\in\partial V$ and suppose $g\in G$ is such that $gH_{n}(a)=H_{n}(b)$, i.e. $gg^{a}_{n}T^{a}_{n}=g^{b}_{n}T^{b}_{n}$. Since the vertices in both sets are in the same orbit, $g^{a}_{n}v^{a}_{n}$ and $g^{b}_{n}v^{b}_{n}$ are elements of $\tilde{V}$ which are in the same orbit, so they are equal, say to some $v\in\tilde{V}$. It suffices to show that $d(v,gv)\leq 6\delta$, since then we can take choose any $K\in\omega$ larger than $\max_{v\in\tilde{V}}|\{g:d(v,gv)\leq 6\delta\}|$.

[TABLE]

Note that since $T^{a}_{n}$ and $T^{b}_{n}$ are infinite, we have that $gg^{a}_{n}a=g^{b}_{n}b$, which we will call $c\in\partial X$. Let $\eta$ be a geodesic from $gg^{a}_{n}v_{0}$ to $c$ with $\eta_{m_{1}}=gv$. Now $v\in gg^{a}_{n}T^{a}_{n}$, so there is some $m_{2}$ with $d(v,\eta_{m_{2}})\leq 2\delta$. Note that by choice of $v^{a}_{n}$, we have $m_{2}\geq m_{1}$. Now let $\gamma$ be a geodesic from $g^{b}_{n}\hat{x}$ to $c$ with $\gamma_{m_{3}}=gv$. By the choice of $v^{b}_{n}$, there is some $m_{4}\leq m_{3}$ such that $d(v,\gamma_{m_{4}})\leq 2\delta$. Also $\eta$ and $\gamma$ are $2\delta$-close after they go through $gv$, so since $m_{2}\geq m_{1}$, there is some $m_{5}\geq m_{3}$ such that $d(\eta_{m_{2}},\gamma_{m_{5}})\leq 2\delta$. Thus

[TABLE]

where the first equality follows from the fact that $\gamma$ is a geodesic. ∎

Now $\operatorname{im}H_{n}$ is analytic, so by Lemma 4.1, there is a Borel equivalence relation $E^{\prime}_{n}$ on $2^{V}$ containing $E_{n}$ whose classes are of size at most $K$. Let $f_{n}:2^{V}\to 2^{\mathbb{N}}$ be a reduction for $E^{\prime}_{n}\leq_{B}\operatorname{id}_{2^{\mathbb{N}}}$, and define $f:\partial V\to(2^{\mathbb{N}})^{\mathbb{N}}$ by $f(a)=(f_{n}(H_{n}(a)):n\in\mathbb{N})$. Write $E^{\prime}$ for the pullback of $E_{1}$ via $f$. Note that since each $E_{n}^{\prime}$ is finite, the relation $E^{\prime}$ is countable. As $E^{\prime}$ is clearly hypersmooth, we get that $E^{\prime}$ is hyperfinite by [Gao09, Theorem 8.1.5]. Now, $f$ is a homomorphism from $E$ to $E_{1}$. Indeed, if $a,b\in\partial V$ with $aEb$, then by Lemma 1.3, there is $N\in\mathbb{N}$ such that $H_{n}(a)E_{n}H_{n}(b)$ for $n\geq N$, and thus $f(x)E_{1}f(y)$. Thus, $E\subseteq E^{\prime}$ is a subrelation of a hyperfinite one, and hence it hyperfinite as well. ∎

Proof of Theorem 1.1.

Let $G$ be a cubulated $\delta$-hyperbolic group. Since hyperfiniteness passes to finite-index extensions [JKL02, Proposition 1.3], by Lemma 5.1, we can assume that $G$ acts freely and cocompactly on a $\operatorname{CAT}(0)$ cube complex $X$. Let $V=X^{(0)}$ be the set of vertices of $X$. Now the statement follows from Theorem 1.4.

∎