Proper isometric actions of hyperbolic groups on $L^p$-spaces

TL;DR

This paper demonstrates that non-elementary hyperbolic groups can act properly and isometrically on certain $L^p$-spaces, extending understanding of their geometric and analytical properties.

Contribution

It constructs proper affine isometric actions of hyperbolic groups on $L^p$-spaces using boundary measures and cocycles, a novel approach in this context.

Findings

Hyperbolic groups act properly on $L^p$-spaces for large $p$

Construction uses boundary measures similar to Bowen-Margulis measure

Hyperbolic groups act on their first $ ext{l}^p$-cohomology group for large $p$

Abstract

We show that every non-elementary hyperbolic group $\G$ admits a proper affine isometric action on $L^p(\bd\G\times \bd\G)$, where $\bd\G$ denotes the boundary of $\G$ and $p$ is large enough. Our construction involves a $\G$-invariant measure on $\bd\G\times \bd\G$ analogous to the Bowen - Margulis measure from the CAT$(-1)$ setting, as well as a geometric cocycle \`a la Busemann. We also deduce that $\G$ admits a proper affine isometric action on the first $\ell^p$-cohomology group $H^1_{(p)}(\G)$ for large enough $p$.