Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

Abstract

We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some $L^p$-space, with $1<p<\infty$, for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated groups. In particular, the Hilbert compression exponent of these groups is equal to 1. This also provides new and optimal estimates for the compression of a uniform embedding of the infinite 3-regular tree into some $L^p$-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable connected Lie groups on $L^p$-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the $L^p$-isoperimetric profile inside balls. We compute the asymptotic behavior of this profile for all amenable connected Lie groups and for all $1\le p<\infty$, providing new geometric invariants of these groups. We also relate the Hilbert compression exponent with other asymptotic quantities such as volume growth and probability of return of random walks.