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

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