A quasi-isometric embedding theorem for 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, Følner functions, and elementary classes of amenable groups.

References in corpus (2)

• A finitely-generated amenable group with very poor compression into Lebesgue spaces
• Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory

Cited by in corpus (8)

• Speed of random walks, isoperimetry and compression of finitely generated groups
• Følner functions and the generic Word Problem for finitely generated amenable groups
• Group Embeddings with Algorithmic Properties
• Computability of Følner sets
• Quantitative nonlinear embeddings into Lebesgue sequence spaces
• Group approximation in Cayley topology and coarse geometry, Part I: Coarse embeddings of amenable groups
• The Poisson boundary of lampshuffler groups
• Algebras and semigroups of locally subexponential growth