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 \`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 $p$.Comment: 17 page