## Duke Mathematical Journal

### 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 (resp., solvable, satisfies a nontrivial identity, elementary amenable, of finite decomposition complexity) whenever $H$ also shares those conditions. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.

#### Article information

Source
Duke Math. J., Volume 162, Number 9 (2013), 1621-1648.

Dates
First available in Project Euclid: 11 June 2013

https://projecteuclid.org/euclid.dmj/1370955541

Digital Object Identifier
doi:10.1215/00127094-2266251

Mathematical Reviews number (MathSciNet)
MR3079257

Zentralblatt MATH identifier
1331.20054

#### Citation

Olshanskii, Alexander Yu.; Osin, Denis V. A quasi-isometric embedding theorem for groups. Duke Math. J. 162 (2013), no. 9, 1621--1648. doi:10.1215/00127094-2266251. https://projecteuclid.org/euclid.dmj/1370955541

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