Proper isometric actions of hyperbolic groups on Lp-spaces

Abstract

We show that every non-elementary hyperbolic group [formula omitted] admits a proper affine isometric action on [formula omitted], where [formula omitted] denotes the boundary of [formula omitted] and [formula omitted] is large enough. Our construction involves a [formula omitted]-invariant measure on [formula omitted] analogous to the Bowen–Margulis measure from the [formula omitted] setting, as well as a geometric, Busemann-type cocycle. We also deduce that [formula omitted] admits a proper affine isometric action on the first [formula omitted]-cohomology group [formula omitted] for large enough [formula omitted]. © 2013, London Mathematical Society. All rights reserved.