Uniformly Lipschitz Hyperbolic Group Actions on ${\ell }^p$

Abstract

Let $\Gamma$ be a finitely generated hyperbolic group, and let $Q$ denote the conformal dimension of its Gromov boundary $\partial \Gamma$. We prove that for every $p<\frac{Q}{Q-1}$, the group $\Gamma$ admits a proper uniformly Lipschitz affine action on ${\ell }^p$ with compression exponent $1/p$. The ${\ell }^p$-space hosting the action is designed from a Markov chain on a specialized hyperbolic filling of $\partial \Gamma$.

arXiv comment26 pages