Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

TL;DR

This paper constructs finitely generated groups with any prescribed Hilbert space compression between 0 and 1, and shows that for many Banach spaces, their compression matches the Hilbert space compression, providing new examples of groups with specific embedding properties.

Contribution

It introduces a method to construct groups with arbitrary Hilbert space compression and demonstrates that their Banach space compression coincides with Hilbert space compression for many spaces.

Findings

Constructed groups with any Hilbert space compression in [0,1]

Groups have asymptotic dimension at most 3 and are exact

First examples of groups with compression 0 in Hilbert and Banach spaces

Abstract

We construct finitely generated groups with arbitrary prescribed Hilbert space compression \alpha from the interval [0,1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.