Groups that (do not) act isometrically on hyperbolic spaces

TL;DR

This paper explores the relationship between group actions on hyperbolic spaces and affine actions on L^p spaces, revealing conditions under which orbits are bounded or unbounded.

Contribution

It establishes a connection between non-elementary isometric actions on hyperbolic spaces and affine actions on L^p spaces, providing new insights into group actions.

Findings

Groups acting non-elementarily on hyperbolic spaces admit affine L^p actions with unbounded orbits for large p.

Groups with fixed point property F_infinity have bounded orbits when acting on hyperbolic spaces.

The results link geometric group actions with functional analysis properties of L^p spaces.

Abstract

In this paper, we show that if a group acts isometrically on a good hyperbolic space of finite volume entropy through a non-elementary action, then it admits an affine action on some $L^p$ -space with an unbounded orbit for sufficiently large $p$. As an application, we prove that any isometric action of a group with the fixed point property $F_\infty$ on a good hyperbolic space must have a bounded orbit.