Two applications of strong hyperbolicity

TL;DR

This paper explores two analytic applications of strongly hyperbolic metrics on hyperbolic groups, including constructing a natural time flow with a unique KMS state and providing a new proof of proper isometric actions on ℓ^p-spaces.

Contribution

It introduces novel applications of strong hyperbolicity, including a natural time flow with a unique KMS state and a simplified proof of proper isometric actions on ℓ^p-spaces.

Findings

Construction of a natural time flow with a KMS state from the boundary measure.

Uniqueness of the KMS state for torsion-free hyperbolic groups.

A new proof that hyperbolic groups act properly on ℓ^p-spaces for large p.

Abstract

We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed-product C*-algebra defined by the action of a hyperbolic group on its boundary. We construct a natural time flow, involving the Busemann cocycle on the boundary. This flow has a natural KMS state, coming from the Hausdorff measure on the boundary, which is furthermore unique when the group is torsion-free. The second application is a short new proof of the fact that a hyperbolic group admits a proper isometric action on an $\ell^p$-space, for large enough $p$.

Equations

$$\displaystyle\langle x,y\rangle_{o}=\tfrac{1}{2}\big{(}|o,x|+|o,y|-|x,y|\big{)}.$$

$$e^{-⟨x,y{⟩}_o}\le e^{-⟨x,z{⟩}_o}+e^{-⟨z,y{⟩}_o}$$

$$\displaystyle\sigma^{c}_{t}\big{(}\sum\phi_{g}g\big{)}=\sum e^{itc(g)}\phi_{g}g$$

$$\omega (b{\sigma }_{i\beta }(a))=\omega (ab)$$

$$\frac{d(g_{\ast }\mu )}{d\mu }=e^{\beta c(g)}$$

$$\displaystyle\omega_{\mu}\big{(}\sum\phi_{g}g\big{)}=\int\phi_{1}\>d\mu$$

$$b(g)(x)=2⟨g,x⟩-|g|(x\in \Gamma )$$

$$b(g)(\xi )=2⟨g,\xi ⟩-|g|(\xi \in \partial \Gamma ).$$

$$d({\xi }_1,{\xi }_2)=e^{-⟨{\xi }_1,{\xi }_2⟩}$$

$$-2⟨gx,gy⟩=b(g^{-1})(x)+b(g^{-1})(y)-2⟨x,y⟩$$

$$d(g\xi ,g\eta {)}^2=e^{b(g^{-1})(\xi )}{e}^{b(g^{-1})(\eta )}d(\xi ,\eta {)}^2$$

$$\frac{d(g^{\ast }\mu )}{d\mu }=e^{Db(g^{-1})}$$

$$b(g)(g^n)=|g^n|-|g^{n-1}|,b(g)(g^{-n})=|g^{-n}|-|g^{-(n+1)}|=-b(g)(g^{n+1}).$$

$$b(g)(g^{-})=-b(g)(g^{+}),$$

$$\displaystyle b(g)(g^{+})=\lim_{n\to\infty}\big{(}|g^{n}|-|g^{n-1}|\big{)}=\lim_{n\to\infty}\frac{|g^{n}|}{n}$$

$$c_g=\sum _{e\in \{1\to g\}}{\delta }_e-\sum _{e\in \{g\to 1\}}{\delta }_e$$

$$\Vert c_g{\Vert }_2^2=2|g|.$$

$$†$$

$$\displaystyle c_{g}(x,y)=\tfrac{1}{2}\big{(}|x|-|g^{-1}x|\big{)}-\tfrac{1}{2}\big{(}|y|-|g^{-1}y|\big{)}$$

$$\displaystyle\Delta=\big{\{}(x,y)\in\Gamma\times\Gamma:K-C\leq|x,y|\leq K+C\big{\}}$$

$$\displaystyle\big{|}e^{-\langle g,x\rangle}-e^{-\langle g,y\rangle}\big{|}\geq e^{-\max\{\langle g,x\rangle,\langle g,y\rangle\}}\>|\langle g,x\rangle-\langle g,y\rangle|.$$

$$|c_g(x,y)|\le e^{|g|}{e}^{-⟨x,y⟩}.$$

$$\sum _{(x,y)\in \Delta }{e}^{-p⟨x,y⟩}\le C_1\sum _{(x,y)\in \Delta }{e}^{-p|x|}\le C_2\sum _{x\in \Gamma }{e}^{-p|x|}$$

$$\Vert c_g{\Vert }_p^p\ge C^{\prime }|g|-C^{\prime \prime }$$

$$\displaystyle\tfrac{1}{2}\big{(}|\gamma(a),\gamma(t_{i})|-|\gamma(b),\gamma(t_{i})|\big{)}-\tfrac{1}{2}\big{(}|\gamma(a),\gamma(t_{i+1})|-|\gamma(b),\gamma(t_{i+1})|\big{)}$$

$$\displaystyle\tfrac{1}{2}\big{(}|a-t_{i}|-|b-t_{i}|\big{)}-\tfrac{1}{2}\big{(}|a-t_{i+1}|-|b-t_{i+1}|\big{)}=t_{i+1}-t_{i}=K.$$

$$\displaystyle\|c_{g}\|^{p}_{p}=\sum_{(x,y)\in\Delta}\big{|}c_{g}(x,y)\big{|}^{p}\geq\sum_{i=0}^{n-1}\big{|}c_{g}(\gamma(t_{i}),\gamma(t_{i+1}))\big{|}^{p}\geq(K-2C)^{p}n.$$

Full text

1. Introduction

Hyperbolicity, in the sense of Gromov, is a coarse notion of negative curvature for metric spaces. In turn, a hyperbolic group is a group which admits a proper and cocompact isometric action on a geodesic hyperbolic space. Such a space is said to be a geometric model for the group. Hyperbolic groups form a large class of groups, and they have received a lot of attention–usually from an algebraic and geometric perspective. Herein, the aims are mostly analytic.

A sharp notion of negative curvature for metric spaces is captured by the CAT$(-1)$ condition. This condition implies, and predates, hyperbolicity. Gromov’s Jugendtraum [6, p.193], that every hyperbolic group admits a geometric model which is CAT$(-1)$, is still wildly open. It is expected to fail, but no counterexamples are known. Let us mention, however, that the past decade has seen great strides in the CAT$(0)$ direction. We now understand that an extraordinary number of hyperbolic groups act on CAT$(0)$ cube complexes.

The search for enhanced geometric models of hyperbolic groups is often motivated by analytic needs. We use the term ‘enhanced hyperbolicity’ as a broad and informal way of describing hyperbolicity with additional CAT$(-1)$ properties. Such desirable properties depend on the specific context. In [13], we introduced the metric notion of strong hyperbolicity. We find this idea satisfactory on two accounts. Firstly, it is an intermediate metric notion between the CAT$(-1)$ condition and hyperbolicity, which grants the additional CAT$(-1)$ properties that, so far, have come up in analytic applications. Secondly, it turns out that every hyperbolic group admits a geometric model which is strongly hyperbolic. We briefly discuss strong hyperbolicity in Section 2 below, and we refer to [13] for more details.

The purpose of this note is to further illustrate the use of strong hyperbolicity in studying analytic aspects of hyperbolic groups. The first application concerns the C∗-crossed product $C(\partial\Gamma)\rtimes\Gamma$ defined by the action of a hyperbolic group $\Gamma$ on its boundary $\partial\Gamma$. We use strong hyperbolicity to construct a natural $\mathbb{R}$-flow on $C(\partial\Gamma)\rtimes\Gamma$, from the Busemann cocycle on the boundary. We show that the Hausdorff measure on the boundary defines a KMS state for this Busemann flow, with inverse temperature equal to the Hausdorff dimension of the boundary. Furthermore, this is the unique KMS state for the flow when $\Gamma$ is torsion-free. Previously, these facts were known in two particular cases: for free groups [5], respectively for uniform lattices in $\mathrm{SO}(n,1)$ [10]. Compare also [8].

The second application is a short new proof of the fact that a hyperbolic group admits a proper isometric action on an $\ell^{p}$-space, for large enough $p\in[1,\infty)$. This result is due to Yu [15], and different proofs have been subsequently offered in [3, 12, 1]. The argument explained in Section 4 provides a link between Haagerup’s original construction for free groups [7], and the boundary construction of [12].

2. Strong hyperbolicity

2.1. Strongly hyperbolic spaces

Let $X$ be a metric space. We write $|x,y|$ for the distance between two points $x,y\in X$. Recall that the Gromov product with respect to a basepoint $o$ is defined by the formula

[TABLE]

The metric space $X$ is said to be strongly hyperbolic if the Gromov product satisfies

[TABLE]

for all $x,y,z,o\in X$. (Compare with the original definition [13, Def.4.1], involving an additional ‘visual’ parameter $\epsilon>0$. The present definition is the normalized case when $\epsilon=1$, and this can always be achieved by rescaling the metric.)

It is easily checked that a strongly hyperbolic space is, in particular, hyperbolic in the usual, Gromov sense. On the other hand, a CAT$(-1)$ space is strongly hyperbolic [13, Thm.5.1]. To put it differently, strong hyperbolicity is a weak CAT$(-1)$ condition. For the purposes of this paper, a useful consequence of strong hyperbolicity is the following [13, Thm.4.2]:

Theorem 2.1**.**

Let $X$ be a strongly hyperbolic space, and let $o\in X$ be a basepoint. Then the Gromov product $\langle\cdot,\cdot\rangle_{o}$ extends continuously to the bordification $X\cup\partial X$, and $e^{-\langle\cdot,\cdot\rangle_{o}}$ is a compatible metric on the boundary $\partial X$.

2.2. Strongly hyperbolic metrics for hyperbolic groups

Let $\Gamma$ be a hyperbolic group. To avoid trivialities, we will always assume that $\Gamma$ is non-elementary. A metric on $\Gamma$ is said to be admissible if it enjoys the following properties:

• (i)

it is equivariant: $|gx,gy|=|x,y|$ for all $g,x,y\in\Gamma$;

• (ii)

it is roughly geodesic: there is a constant $C\geq 0$, so that for every pair of points $x,y\in\Gamma$ there is a (not necessarily continuous) map $\gamma:[a,b]\to\Gamma$ satisfying $\gamma(a)=x$, $\gamma(b)=y$, and $|s-t|-C\leq|\gamma(s),\gamma(t)|\leq|s-t|+C$ for all $s,t\in[a,b]$;

• (iii)

it is quasi-isometric to any word metric on $\Gamma$.

An admissible metric on $\Gamma$ is hyperbolic, since hyperbolicity is a quasi-isometry invariant for roughly geodesic spaces.

Admissible metrics naturally arise from geometric models for $\Gamma$. Let $X$ be a geodesic hyperbolic space on which $\Gamma$ acts isometrically, properly and cocompactly, and pick a basepoint $o\in X$. Then the orbit metric on $\Gamma$, given by $|g,h|_{o}:=|go,ho|$, is admissible. (An innocuous issue is that $o$ might have non-trivial stabilizer. This is easily made irrelevant either by language, allowing pseudo-metrics instead of metrics, or by coarse bookkeeping.)

If $\Gamma$ admits a CAT$(-1)$ geometric model, then the induced orbit metrics on $\Gamma$ are strongly hyperbolic. The following theorem is a general statement to that effect, circumventing the delicate question whether a CAT$(-1)$ geometric model is always available.

Theorem 2.2**.**

There exist admissible metrics on $\Gamma$ which are strongly hyperbolic.

Implicitly, this was first proved in [11] by an involved construction of combinatorial flavour. In [13] we show that there are, in fact, natural admissible metrics that are strongly hyperbolic. Namely, the Green metric defined by any symmetric and finitely supported random walk on $\Gamma$ is, up to a rescaling, strongly hyperbolic [13, Thm.6.1].

3. The Busemann flow for boundary actions of hyperbolic groups

3.1. Preliminaries

Let us start with some general facts on cocycles, flows, and KMS states for crossed-products. These matters are well-known, and they go back to Renault’s foundational work [14]. A minor difference is that we choose to work with reduced crossed-products, rather than full crossed-products.

Let $G$ be a discrete countable group acting by homeomorphisms on a compact Hausdorff space $\Omega$. The algebraic crossed-product $C(\Omega)\rtimes_{\mathrm{alg}}G$ consists of finite sums of the form $\sum\phi_{g}g$, where $\phi_{g}\in C(\Omega)$ and $g\in G$. This is an algebra for the multiplication whose defining rule is that $(\phi g)(\psi h)=\phi(g.\psi)gh$. The reduced crossed-product $C(\Omega)\rtimes_{\mathrm{r}}G$ is the reduced C∗-completion of $C(\Omega)\rtimes_{\mathrm{alg}}G$.

A flow on a C∗-algebra $A$ is a strongly continuous group homomorphism $\sigma:\mathbb{R}\to\mathrm{Aut}(A)$. On crossed-products, cocycles give rise to flows, as follows. Consider a cocycle $c:G\to C(\Omega,\mathbb{R})$, the real-valued continuous maps on $\Omega$. (Throughout this paper, the cocycle property is in the additive sense: $c(gh)=c(g)+g.c(h)$ for all $g,h\in G$.) Then there is a flow $\sigma^{c}$ on $C(\Omega)\rtimes_{\mathrm{r}}G$, defined by the formula

[TABLE]

on $C(\Omega)\rtimes_{\mathrm{alg}}G$.

Let $\sigma$ be a flow on a C*-algebra $A$, and $\beta\in\mathbb{R}$. A state $\omega$ on $A$ is said to be a $\beta$-KMS state for $\sigma$ if

[TABLE]

for all $a,b$ in a dense subalgebra of $\sigma$-entire elements of $A$. We refrain from defining the notion of $\sigma$-entire elements of $A$, except to mention that the $\sigma$-entire elements form a dense $*$-subalgebra of $A$. The parameter $\beta$ is called inverse temperature.

Now consider the flow $\sigma^{c}$ on $C(\Omega)\rtimes_{\mathrm{r}}G$, induced by a cocycle $c$ as above. Then all elements in $C(\Omega)\rtimes_{\mathrm{alg}}G$ are entire. Let $\omega$ be a $\beta$-KMS state for $\sigma^{c}$. As for any state on $C(\Omega)\rtimes_{\mathrm{r}}G$, the restriction of $\omega$ to $C(\Omega)$ defines a probability $\mu$ on $\Omega$. (Here, and in what follows, we use the term ‘probability’ as a shorthand for ‘regular Borel probability measure’.) The KMS condition means that $\mu$ is $e^{\beta c}$-conformal, in the sense that

[TABLE]

for each $g\in G$. Conversely, let $\mu$ be an $e^{\beta c}$-conformal probability on $\Omega$. Consider the state $\omega_{\mu}$ on $C(\Omega)\rtimes_{\mathrm{r}}G$, defined by

[TABLE]

on $C(\Omega)\rtimes_{\mathrm{alg}}G$. In other words, $\omega_{\mu}$ is the composition of the standard expectation $C(\Omega)\rtimes_{\mathrm{r}}G\to C(\Omega)$ with $\mu$, viewed as a state on $C(\Omega)$. Then $\omega_{\mu}$ is a $\beta$-KMS state for the cocycle flow $\sigma^{c}$.

The following result says that the previous construction is the only source of KMS states for $\sigma^{c}$, whenever $c$ satisfies a certain non-vanishing condition.

Theorem 3.1** (Kumjian - Renault [9]).**

Consider a cocycle flow $\sigma^{c}$ on $C(\Omega)\rtimes_{\mathrm{r}}G$. Assume that, for all non-trivial $g\in G$, $c(g)$ is non-zero at each fixed point of $g$. Then every $\beta$-KMS state for $\sigma^{c}$ is of the form $\omega_{\mu}$ for some $e^{\beta c}$-conformal probability $\mu$ on $\Omega$.

The non-vanishing condition could be thought of as a strong cohomological non-triviality. For if the cocycle $c:G\to C(\Omega,\mathbb{R})$ is of the form $c(g)=g.\theta-\theta$, then $c(g)$ vanishes at each fixed point of $g$, for all $g\in G$.

3.2. The boundary crossed product of a hyperbolic group

Now let $\Gamma$ be a hyperbolic group and consider the reduced crossed-product $C(\partial\Gamma)\rtimes_{\mathrm{r}}\Gamma$, defined by the action of $\Gamma$ on its boundary $\partial\Gamma$. Endow $\Gamma$ with a strongly hyperbolic, admissible metric.

A remarkable cocycle on $\Gamma$ is the Busemann cocycle. To begin, there is the group Busemann cocycle, given by

[TABLE]

for each $g\in\Gamma$. Here, and in all that follows, the Gromov product is based at the identity, and we write $|g|$ for $|1,g|$, the distance from $g$ to the identity. The cocycle property for $b$ is easily checked. In fact, writing $b(g)(x)=|x|-|g^{-1}x|$ exhibits $b$ as a coboundary.

Secondly, and more importantly for the purposes of this section, there is a boundary Busemann cocycle. By Theorem 2.1, the group Busemann cocycle extends, by continuity and as a continuous function, to the boundary. The boundary Busemann cocycle is given, for each $g\in\Gamma$, by

[TABLE]

The boundary Busemann cocycle $b$ takes values in $C(\partial\Gamma,\mathbb{R})$, so it defines a flow $\sigma^{b}$ on $C(\partial\Gamma)\rtimes_{\mathrm{r}}\Gamma$.

On the other hand, by Theorem 2.1, once again, the Gromov product based at the identity induces a compatible metric

[TABLE]

on $\partial\Gamma$. Let $\mu$ be the probability on $\partial\Gamma$ defined by normalizing the Hausdorff measure, and let $D$ denote the Hausdorff dimension of $\partial\Gamma$.

Theorem 3.2**.**

Consider the Busemann cocycle flow $\sigma^{b}$ on $C(\partial\Gamma)\rtimes_{\mathrm{r}}\Gamma$. Then the probability $\mu$ induces a KMS state $\omega_{\mu}$ for $\sigma^{b}$, at inverse temperature $D$. If $\Gamma$ is torsion-free, then $\omega_{\mu}$ is the unique KMS state for $\sigma^{b}$.

Proof.

In order for $\omega_{\mu}$ to be a KMS state for $\sigma^{b}$ at inverse temperature $D$, we need to know that the probability $\mu$ is $e^{Db}$-conformal. Fix $g\in\Gamma$. We have

[TABLE]

for all $x,y\in\Gamma$. This identity extends by continuity to the boundary, leading to

[TABLE]

for all $\xi,\eta\in\partial\Gamma$. It follows, see [12, Lem.8], that

[TABLE]

for each $g\in G$. Up to replacing $g$ by $g^{-1}$, this is means that $\mu$ is $e^{Db}$-conformal, as desired.

Now let us turn to the uniqueness statement, in which $\Gamma$ is assumed to be torsion-free. We wish to apply the Kumjian - Renault criterion, so let us check that $b$ satisfies the non-vanishing condition of Theorem 3.1. Let $g$ be a non-trivial element of $\Gamma$. Then the following properties hold. Firstly, the infinite cyclic subgroup generated by $g$ is quasi-isometrically embedded in $\Gamma$. Secondly, there are two distinct points $g^{+},g^{-}\in\partial\Gamma$ such that $g^{n}\to g^{+}$ and $g^{-n}\to g^{-}$ as $n\to\infty$. Thirdly, the points fixed by $g$ on the boundary are precisely $g^{+}$ and $g^{-}$.

For the group Busemann cocycle, we have

[TABLE]

Letting $n\to\infty$, the second relation yields

[TABLE]

while the first leads to

[TABLE]

by the discrete l’Hospital rule. But the right-hand limit is positive, as $g$ is undistorted, and we conclude that $b(g)(g^{+})>0$ and $b(g)(g^{-})<0$.

We deduce that a KMS state for $\sigma^{b}$ at inverse temperature $D^{\prime}$ must be induced by a probability $\mu^{\prime}$ on $\partial\Gamma$ which is $e^{D^{\prime}b}$-conformal. Results of Coornaert [4], and their generalizations to the roughly geodesic context by Blachère, Haïssinsky, and Mathieu [2], imply that $D^{\prime}=D$ and $\mu^{\prime}=\mu$. ∎

4. The Haagerup cocycle for hyperbolic groups

4.1. The Haagerup cocycle for free groups

Let $\mathbb{F}$ be a non-abelian free group. Then $\mathbb{F}$ admits a proper isometric action on a Hilbert space. This is due to Haagerup [7], up to a slight reinterpretation, and his elegant construction runs as follows.

Consider the standard Cayley graph of $\mathbb{F}$ with respect to the free generators and their inverses. This is a regular undirected tree. Let $\vec{E}$ be the set of its oriented edges. Then $\mathbb{F}$ acts on $\vec{E}$ in a natural way, and we may consider the corresponding orthogonal representation of $\mathbb{F}$ on $\ell^{2}(\vec{E})$. Next, we perturb this linear isometric action by a cocycle $c:\mathbb{F}\to\ell^{2}(\vec{E})$. Given $g\in\mathbb{F}$, let $c_{g}$ be the following function on $\vec{E}$: $c_{g}$ is supported on the geodesic path joining $g$ to the identity $1$, and for an oriented edge $e$ lying on this path we value $c_{g}(e)$ to be $+1$ or $-1$ according to whether $e$ points towards or away from $g$. In short:

[TABLE]

The cocycle property, $c_{gh}=c_{g}+g.c_{h}$ for all $g,h\in\mathbb{F}$, can be seen by drawing the geodesic tripod defined by $1$, $g$, and $gh$, and noting that the oriented edges lying on the leg towards $g$ cancel out. Clearly, $c_{g}\in\ell^{2}(\vec{E})$ and

[TABLE]

In particular, the cocycle $c$ is proper: $\|c_{g}\|_{2}\to\infty$ as $g\to\infty$ in $\mathbb{F}$. It follows that the affine isometric action of $\mathbb{F}$ on $\ell^{2}(\vec{E})$ given by $(g,\phi)\mapsto g.\phi+c_{g}$ is proper. Note that this construction applies, in fact, to any space $\ell^{p}(\vec{E})$ for $p\in[1,\infty)$.

We wish to adapt Haagerup’s construction to a general hyperbolic context, and we start by recasting the above cocycle in a more convenient form. Firstly, we think of the oriented edge-set $\vec{E}$ as the set $\{(x,y)\in\mathbb{F}\times\mathbb{F}:|x,y|=1\}$. Secondly, we note that the cocycle $c$ can be described by in metric terms by the following formula:

[TABLE]

Recall that $\langle\cdot,\cdot\rangle$ denotes the Gromov product based at the identity. In this form, the cocycle property is even more transparent: writing

[TABLE]

we obtain the coboundary formula $c_{g}=F-g.F$, for F(x,y)=\tfrac{1}{2}\big{(}|x|-|y|\big{)}.

4.2. The Haagerup cocycle for hyperbolic groups

Let $\Gamma$ be a hyperbolic group, which we may assume to be non-elementary. Endow $\Gamma$ with a strongly hyperbolic admissible metric. We also consider a coarse relative of the underlying set we have used in the free group case. Namely, let

[TABLE]

where $C\geq 0$ is a rough geodesic constant, and $K>0$ is another constant. For the purposes of the following theorem, we ask that $K>2C$. Note that $\Delta$ is non-empty. This can be seen by choosing a convenient point along a rough geodesic from the identity to some sufficiently remote group element.

The group $\Gamma$ acts on $\Delta$, by $g.(x,y)=(gx,gy)$. Let $c_{g}$ be defined on $\Delta$ by the metric formula ($\dagger$ ‣ 4.1). Then $c$ is a cocycle for $\Gamma$, for the same reasons as explained above.

Theorem 4.1**.**

For large enough $p\in[1,\infty)$, the affine isometric action of $\Gamma$ on $\ell^{p}(\Delta)$ given by $(g,\phi)\mapsto g.\phi+c_{g}$ is well-defined and proper.

Proof.

For the action to be well-defined, we need to have $c_{g}\in\ell^{p}(\Delta)$ for each $g\in\Gamma$. An application of the mean value theorem to the function $t\mapsto e^{-t}$ yields

[TABLE]

The left-hand side is at most $e^{-\langle x,y\rangle}$, thanks to strong hyperbolicity. On the right-hand side, both $\langle g,x\rangle$ and $\langle g,y\rangle$ are at most $|g|$. It follows that

[TABLE]

We complete the argument by showing that $e^{-\langle\cdot,\cdot\rangle}\in\ell^{p}(\Delta)$ for large enough $p\in[1,\infty)$. If $(x,y)\in\Delta$ then $\langle x,y\rangle\geq|x|-|x,y|\geq|x|-(K+C)$. We deduce that

[TABLE]

and the latter sum converges when $p$ is large enough.

For the action to be proper, we need to argue that $\|c_{g}\|_{p}\to\infty$ as $g\to\infty$ in $\Gamma$. In fact, we show that there are constants $C^{\prime},C^{\prime\prime}>0$, depending only on $K$, $C$, and $p$, such that

[TABLE]

for each $g\in\Gamma$.

Let $\gamma:[a,b]\to\Gamma$ be a rough geodesic joining the identity to $g$. The basic idea is that $|c_{g}(x,y)|$ is roughly $|x,y|$ whenever $x$ and $y$ lie on $\gamma$, and that we can find about $|g|/K$ pairs of points on $\gamma$ that belong to $\Delta$. Now let us be precise.

Consider the elements $\gamma(t_{i})\in\Gamma$ arising from a partition $a=t_{0}<\ldots<t_{n}\leq b$ into $n$ intervals of length $K$, and a remainder of length less than $K$. Then $|\gamma(t_{i}),\gamma(t_{i+1})|$ is within $C$ of $|t_{i}-t_{i+1}|=K$, so $(\gamma(t_{i}),\gamma(t_{i+1}))\in\Delta$. Also, $c_{g}(\gamma(t_{i}),\gamma(t_{i+1}))$ can be written as

[TABLE]

which is within $2C$ of

[TABLE]

In particular, $c_{g}(\gamma(t_{i}),\gamma(t_{i+1}))\geq K-2C>0$, according to our assumption on $K$. Hence

[TABLE]

On the other hand, we can relate $n$ and $|g|$. The way we defined the partition implies that $K(n+1)>b-a$, and $b-a\geq|g|-C$ by using the rough geodesic property at the endpoints. Therefore n\geq\big{(}|g|-(K+C)\big{)}/K, and the desired claim follows. ∎

We end by pointing out that the cocycle used in [12] is the boundary analogue of ($\dagger$ ‣ 4.1), namely $c_{g}(\xi,\eta)=\langle g,\xi\rangle-\langle g,\eta\rangle$ for $\xi,\eta\in\partial\Gamma$.

Acknowledgments**.**

I thank Jean Renault for discussions around §3.1.