Bi-exactness of relatively hyperbolic groups

In this section, we introduce some equivalent conditions of bi-exact groups. The definition of an array given below was suggested in [4].

Conditions (2) and (3) in Proposition 2.2 are simplified versions of Proposition 2.3 of [4] and Proposition 2.7 of [19] respectively.

There are several equivalent definitions of relatively hyperbolic groups due to Gromov, Bowditch, Farb, and Osin. We adopt Farb’s definition with the Bounded Coset Penetration property replaced by fineness of coned-off Cayley graphs. [7], [5, Appendix], and [11, Appendix] contain proofs of the equivalence of these definitions. To state our definition, we first prepare some terminologies. When we consider a graph $Y$ without loops or multiple edges, the edge set $E(Y)$ is defined as a irreflexive symmetric subset of $V(Y)\times V(Y)$, where $V(Y)$ is the vertex set (see Section 2.3 for details).

We set the length of any edge of $\widehat{\Gamma}$ to be 1. $G$ acts on $\widehat{\Gamma}$ naturally by graph automorphism. For a vertex $v\in V(\widehat{\Gamma})$, we denote its stabilizer by $G_{v}=\{g\in G\mid gv=v\}$.

The following is the definition of relatively hyperbolic groups on which we work. The reader is referred to [5, Appendix] for the equivalence of Definition 2.7 and the other definitions.

Some definitions (e.g. Osin’s definition in [11]) can be stated without requiring $\Lambda$ to be finite. In Osin’s definition, finiteness of $\Lambda$ follows if $G$ is finitely generated.

In [10], Mineyev and Yaman constructed a bicombing of a fine hyperbolic graph using the same idea as in [9]. We first discuss 1-chains on a graph and a unitary representation of a group acting on a graph in general and then introduce Mineyev-Yaman’s bicombing.

In this section, we consider graphs without loops nor multiple edges, following [10] and [1]. More precisely, for a graph $Y$ with a vertex set $V(Y)$, its edge set $E(Y)$ is a subset of $V(Y)\times V(Y)\setminus\{(v,v)\mid v\in V(Y)\}$ such that $\overline{E(Y)}=E(Y)$, where we define $\overline{(u,v)}=(v,u)$ for any $(u,v)\in V(Y)^{2}$. We call a subset $E^{+}(V)$ of $E(Y)$ a set of positive edges if it satisfies $E(V)=E^{+}(V)\sqcup\overline{E^{+}(V)}$. Choosing a set of positive edges $E^{+}(V)$ means choosing directions of edges. Note that when we consider a group action on a graph, we allow inversion of edges. We begin with auxiliary notations.

For a graph $Y$, we denote by $C_{1}(Y)$ the set of 1-chains on $Y$ over $\mathbb{C}$. More precisely, $C_{1}(Y)$ is defined as follows. Consider a direct product $\mathbb{C}^{E(Y)}=\Big{\{}\sum_{e\in E(Y)}c_{e}e\;\Big{|}\;c_{e}\in\mathbb{C}\Big{\}}$ of vector spaces over $\mathbb{C}$. Note that this summation notation is just a formal sum. We define a quotient space $\ell^{0}(Y)$ by

$$\ell^{0}(Y)=\mathbb{C}^{E(Y)}\;\Big{/}\;\Big{\{}\sum_{e\in E(Y)}c_{e}e\in\mathbb{C}^{E(Y)}\;\Big{|}\;c_{e}=c_{\bar{e}}\;\forall e\in E(Y)\Big{\}}.$$

Here, we denote by $[e]=[u,v]$ the element in $\ell^{0}(Y)$ corresponding to $e=(u,v)\in\mathbb{C}^{E(Y)}$ where $e=(u,v)\in E(Y)$. Note that for any $e=(u,v)\in E(Y)$, we have

$$[\bar{e}]=[v,u]=-[u,v]=-[e].$$

Fix a set of positive edges $E^{+}(Y)$. The map

$$\mathbb{C}^{E^{+}(Y)}\ni\sum_{e\in E^{+}(Y)}c_{e}e\mapsto\sum_{e\in E^{+}(Y)}c_{e}[e]\in\ell^{0}(Y)$$

is an isomorphism of vector spaces. The space of 1-chains is defined by

$$C_{1}(Y)=\Big{\{}\sum_{e\in E^{+}(Y)}c_{e}[e]\in\ell^{0}(Y)\;\Big{|}\;\exists F\subset E^{+}(Y),\;|F|<\infty\;\wedge\;c_{e}=0\;\forall e\notin F\Big{\}}.$$

Note that $C_{1}(Y)$ is independent of the choice of $E^{+}(Y)$. For $a,b\in V(Y)$ and a path $q=(v_{0},v_{1},\cdots,v_{n})$ from $a=v_{0}$ to $b=v_{n}$, we define a 1-chain

$$q=[v_{0},v_{1}]+\cdots+[v_{n-1},v_{n}]\in C_{1}(Y).$$

(2)

Here, we use the same notation $q$ to denote a path and the corresponding 1-chain by abuse of notation.

For each $p\in[1,\infty)$ (in fact, we use only cases $p=1,2$), define a subspace $\ell^{p}(Y)$ of $\ell^{0}(Y)$ by

$$\ell^{p}(Y)=\Big{\{}\sum_{e\in E^{+}(Y)}c_{e}[e]\in\ell^{0}(Y)\;\Big{|}\;\sum_{e\in E^{+}(Y)}|c_{e}|^{p}<\infty\Big{\}}$$

and for each $\xi=\sum_{e\in E^{+}(Y)}c_{e}[e]\in\ell^{p}(Y)$ define a norm

$$\|\xi\|_{p}=\left(\sum_{e\in E^{+}(Y)}|c_{e}|^{p}\right)^{1/p}.$$

$(\ell^{p}(Y),\|\cdot\|_{p})$ becomes a Banach space isomorphic to $\ell^{p}(E^{+}(Y))$. In particular, $(\ell^{2}(Y),\|\cdot\|_{2})$ is a Hilbert space. Note that $(\ell^{p}(Y),\|\cdot\|_{p})$ doesn’t depend on the choice of $E^{+}(Y)$. We also have $C_{1}(Y)\subset\ell^{p}(Y)$ for any $p\in[1,\infty)$.

There exists a unique well-defined linear map

$$\partial\colon C_{1}(Y)\to C_{0}(Y)=\Big{\{}\sum_{v\in F}c_{v}v\mid F\subset V(Y),\;|F|<\infty,\;c_{v}\in\mathbb{C}\Big{\}}$$

such that $\partial[u,v]=v-u$ for any $(u,v)\in E(Y)$.

Theorem 2.11 is a simplified version of Theorem 47 and Proposition 46 (3) in [10].

In this section, we define Hull-Osin’s separating cosets and explain their properties required for our discussion in Section 3.3. The notion of separating cosets of hyperbolically embedded subgroups was first introduced by Hull and Osin in [8] and further developed by Osin in [12]. There is one subtle difference in the definition of separationg cosets in [8] and in [12], which is explained in Remark 2.21, though other terminologies and related propositions are mostly the same between them. With regards to this difference, we follow definitions and notations of [12] in our discussion. In this section, suppose that $G$ is a group, $X$ is a subset of $G$, and $\{H_{\lambda}\}_{\lambda\in\Lambda}$ is a collection of subgroups of $G$ such that $X\cup(\bigcup_{\lambda\in\Lambda}H_{\lambda})$ generates $G$. Note that $\Lambda$ and $X$ are possibly infinite. We begin with defining some auxiliary concepts.

Let $\mathcal{H}=\bigsqcup_{\lambda\in\Lambda}(H_{\lambda}\setminus\{1\})$. Note that this union is disjoint as sets of labels, not as subsets of $G$. Let $\Gamma(G,X\cup\mathcal{H})$ be the Cayley graph of $G$ with respect to $X\sqcup\mathcal{H}$, which allows loops and multiple edges, that is, its vertex set is $G$ and its positive edge set is $G\times(X\sqcup\mathcal{H})$. We call $\Gamma(G,X\cup\mathcal{H})$ the relative Cayley graph.

For each $\lambda\in\Lambda$, we consider the Cayley graph $\Gamma(H_{\lambda},H_{\lambda}\setminus\{1\})$, which is a subgraph of $\Gamma(G,X\cup\mathcal{H})$, and define a metric $\widehat{d_{\lambda}}\colon H_{\lambda}\times H_{\lambda}\to[0,\infty]$ as follows. We say that a path $p$ in $\Gamma(G,X\cup\mathcal{H})$ is $\lambda$-admissible, if $p$ doesn’t contain any edge of $\Gamma(H_{\lambda},H_{\lambda}\setminus\{1\})$. Note that $p$ can contain an edge whose label is an element of $H_{\lambda}$ (e.g. the case when the initial vertex of the edge is not in $H_{\lambda}$) and also $p$ can pass vertices of $\Gamma(H_{\lambda},H_{\lambda}\setminus\{1\})$. For $f,g\in H_{\lambda}$, we define $\widehat{d_{\lambda}}$ to be the minimum of lengths of all $\lambda$-admissible paths from $f$ to $g$. If there is no $\lambda$-admissible path from $f$ to $g$, then we define $\widehat{d_{\lambda}}(f,g)=\infty$. For convenience, we extend $\widehat{d_{\lambda}}$ to $\widehat{d_{\lambda}}\colon G\times G\to[0,\infty]$ by defining $\widehat{d_{\lambda}}(f,g)=\widehat{d_{\lambda}}(1,f^{-1}g)$ if $f^{-1}g\in H_{\lambda}$ and $\widehat{d_{\lambda}}(f,g)=\infty$ otherwise.

The following is a simplified version of Proposition 4.28 of [6].

In the remainder of this section, suppose that $\{H_{\lambda}\}_{\lambda\in\Lambda}$ is hyperbolically embedded in $(G,X)$.

The following proposition is important, which is a particular case of Proposition 4.13 of [6].

In what follows, we fix any constant $D>0$ with

$$D\geq 3C.$$

(3)

We can now define separating cosets.

The following lemma is immediate from the above definition.

We state some nice properties of separating cosets, all of which were proven in [8]. For $f,g\in G$, we denote by $\mathcal{G}(f,g)$ the set of all geodesic paths in $\Gamma(G,X\cup\mathcal{H})$ from $f$ to $g$.

Proof of Lemma 2.23 is the same as Lemma 3.3 of [8], though their statements are slightly different. Actually, in Lemma 2.23, we don’t need to assume $f^{-1}g\notin H_{\lambda}$. This difference comes from the difference of definitions of separating cosets, which was mentioned in Remark 2.21.

The following lemma makes $S_{\lambda}(f,g;D)$ into a totally ordered set.

Next, we define a set of pairs of entrance and exit points of a separating coset. That is, for $f,g\in G$, $\lambda\in\Lambda$, and $xH_{\lambda}\in S_{\lambda}(f,g;D)$, we define

$$E(f,g;xH_{\lambda},D)=\{(p_{in}(xH_{\lambda}),p_{out}(xH_{\lambda}))\mid p\in\mathcal{G}(f,g)\}.$$

Note that because $xH_{\lambda}$ is a $(f,g;D)$-separating coset, any geodesic from $f$ to $g$ penetrates $xH_{\lambda}$ by Lemma 2.23 (a).

The following lemma is crucial in constructing an array that satisfies the bounded area condition.

In this section, we explain the idea of the construction of a proper array in the proof of Proposition 3.20 (see (9) (10)) by considering two particular cases.

First, suppose that $G$ is a hyperbolic group i.e., the collection of peripheral subgroups is empty. We take a symmetric finite generating set $X_{0}$ of $G$ with $1\in X_{0}$ and define

$$X=X_{0}^{2}=\{gh\mid g,h\in X_{0}\}.$$

The Cayley graph $\Gamma(G,X)$, as defined by (1), has no loops or multiple edges and is a 2-vertex-connected locally finite hyperbolic graph. Let $Y$ be the barycentric subdivision of $\Gamma(G,X)$. Note that $Y$ is fine, $G$ acts on $Y$ without inversion of edges, and all edge stabilizers are trivial. By Theorem 2.11, there exists a $G$-equivariant anti-symmetric $\mathbb{Q}$-bicombing $q$ of $Y$ and a constant $T\geq 0$ such that for any $a,b,c\in V(Y)$, we have

$$\|q[a,b]+q[b,c]+q[c,a]\|_{1}\leq T.$$

We define a map $Q\colon G\to\ell^{2}(Y)$ by (see Definition 3.4 for details of this definition)

$$Q(g)=\widetilde{q[1,g]}.$$

It is straightforward to check that $Q$ is an array into $(\ell^{2}(Y),\pi)$ (cf. Definition 2.8) by using Lemma 3.5 (2). Also, we have

$$d_{Y}(1,g)\leq\|q[1,g]\|_{1}=\|Q(g)\|_{2}^{2}$$

(see Remark 2.12, Lemma 3.3, and Lemma 3.5 (1)). Since $Y$ is a barycentric subdivision of $\Gamma(G,X)$, we have $d_{Y}(1,g)=2d_{X}(1,g)$ for any $g\in G$, where $d_{X}$ is a word metric on $G$ with respect to $X$. Hence, for any $N\in\mathbb{N}$, if $g\in G$ satisfies $\|Q(g)\|_{2}\leq N$, we have

$$d_{X}(1,g)=\frac{1}{2}d_{Y}(1,g)\leq\frac{1}{2}\|Q(g)\|_{2}^{2}\leq\frac{1}{2}N^{2}.$$

This implies that $Q$ is proper. Combined with Proposition 2.2 (3), this gives another proof of the fact that hyperbolic groups are bi-exact. Here, note that $(\ell^{2}(Y),\pi)$ is a direct sum of copies $(\ell^{2}(G),\lambda_{G})$, because $G$ acts on $Y$ without inversion of edges and its action on $E(Y)$ is free. Hence, it is weakly contained by $(\ell^{2}(G),\lambda_{G})$. We can also use Mineyev’s bicombing in [9] instead of Theorem 2.11 in this case.

Second, suppose that $G$ is a free product of bi-exact groups $H_{1}$ and $H_{2}$. In this case, $\{H_{1},H_{2}\}$ are peripheral subgroups and we can construct a proper array on $G$ using the normal forms of elements of the free product and proper arrays on $H_{1}$ and $H_{2}$ as follows. Let $r_{i}\colon H_{i}\to\ell^{2}(H_{i})$ be a proper array into the left regular representation $(\ell^{2}(H_{i}),\lambda_{H_{i}})$ for each $i=1,2$. We can assume $\|r_{i}(a)\|_{2}\geq 1$ for any $a\in H_{i}\setminus\{1\}$ by changing values of $r_{i}$ on a finite subset of $H_{i}$, if necessary. Also, we regard each $r_{i}$ as a map from $H_{i}$ to $\ell^{2}(G)$ by composing it with the embedding $\ell^{2}(H_{i})\hookrightarrow\ell^{2}(G)$. For $g\in G\setminus\{1\}$, without loss of generality, let $g=h_{1}k_{1}\cdots h_{n}k_{n}$ be the normal form of $g$, where $h_{1},\cdots,h_{n}\in H_{1}\setminus\{1\}$ and $k_{1},\cdots,k_{n}\in H_{2}\setminus\{1\}$. We define a map $R_{1}\colon G\to\ell^{2}(G)$ by

$$R_{1}(g)=\lambda_{G}(1)r_{1}(h_{1})+\lambda_{G}(h_{1}k_{1})r_{1}(h_{2})+\cdots+\lambda_{G}(h_{1}k_{2}\cdots h_{n-1}k_{n-1})r_{1}(h_{n}).$$

Here, we define $R_{1}(1)=0$. In the same way, we define a map $R_{2}\colon G\to\ell^{2}(G)$ from $r_{2}$. It is not difficult to show that the map $R\colon G\to\ell^{2}(G)\oplus\ell^{2}(G)$ defined by

$$R(g)=(R_{1}(g),R_{2}(g))$$

is a proper array into $(\ell^{2}(G)\oplus\ell^{2}(G),\lambda_{G}\oplus\lambda_{G})$.