Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

TL;DR

This paper studies how groups embed into Lp-spaces, revealing their asymptotic compression behavior, especially for Lie groups and hyperbolic groups, and constructs optimal affine actions with implications for geometric group theory.

Contribution

It characterizes the asymptotic compression rates for a broad class of groups and constructs explicit affine isometric actions with optimal compression, introducing new geometric invariants.

Findings

Hilbert compression rate of these groups equals 1

Provides optimal estimates for embedding the infinite 3-regular tree into Lp-spaces

Computes the Lp-isoperimetric profile for all amenable connected Lie groups

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.