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

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 à 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 $p$.

17 pages

References in corpus (2)

• Property (T) and rigidity for actions on Banach spaces
• Flows and joins of metric spaces

Cited by in corpus (5)

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• Topological flows for hyperbolic groups
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