Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

Abstract

We characterize the asymptotic behaviour of the compression associated to a uniform embedding into some Lp-space for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated groups. In particular, the Hilbert compression rate 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 Lp-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable Lie groups on Lp-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the Lp-isoperimetric profile inside balls. We compute this profile for all amenable connected Lie groups and for all finite p, providing new geometric invariants of these groups. We also relate the Hilbert compression rate with other asymptotic quantities such as volume growth and probability of return of random walks.

38 pages, modification: correct proof of the lower bound 2/3 of the compression of (Z \wr Z)

References in corpus (3)

• A product of trees as universal space for hyperbolic groups
• Embeddings of discrete groups and the speed of random walks
• Compression of uniform embeddings into Hilbert space

Cited by in corpus (5)

• The wreath product of Z with Z has Hilbert compression exponent 2/3
• Vanishing of the first reduced cohomology with values in an $L^p$-representation
• On the L^p-distorsion of finite quotients of amenable groups
• Large scale Sobolev inequalities on metric measure spaces and applications
• Hilbert space compression for free products and HNN-extensions