---
abstract: 'Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group ${\rm Iso}(X)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(\Gamma,\mathbb{R})$, $H_b^2(\Gamma,\ell^p(\Gamma))$ $(1< p <\infty)$ are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct product.'
author:
- Ursula Hamenstädt
date: 'September 27, 2006'
title: Bounded cohomology and isometry groups of hyperbolic spaces
---

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Introduction
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A *Banach module* for a countable group $\Gamma$ is a Banach space $E$ together with a homomorphism of $\Gamma$ into the group of linear isometries of $E$. For every such Banach module $E$ for $\Gamma$ and every $i\geq 1$, the group $\Gamma$ naturally acts on the vector space $L^\infty(\Gamma^i,E)$ of bounded functions $\Gamma^i\to E$. If we denote by $L^\infty(\Gamma^i,E)^\Gamma\subset L^\infty(\Gamma^i,E)$ the linear subspace of all $\Gamma$-invariant such functions, then the *second bounded cohomology group* $H_b^2(\Gamma,E)$ of $\Gamma$ with coefficients $E$ is defined as the second cohomology group of the complex $$0\to
