A quasi-isometric embedding theorem for groups

TL;DR

This paper proves that groups with at most exponential growth can be bi-Lipschitz embedded into finitely generated groups with specific properties, impacting understanding of group embeddings and related geometric group theory concepts.

Contribution

It establishes a quasi-isometric embedding theorem for groups, linking growth conditions to embeddings into finitely generated groups with various algebraic properties.

Findings

Groups of at most exponential growth can be embedded into finitely generated groups with desired properties.

Applications to compression functions of Lipschitz embeddings into Banach spaces.

Insights into F{46}lner functions and elementary classes of amenable groups.

Abstract

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F{\o}lner functions, and elementary classes of amenable groups.