Bilipschitz versus quasi-isometric equivalence for higher rank lamplighter groups

TL;DR

This paper constructs higher rank lamplighter groups that are quasi-isometric but not bilipschitz equivalent, extending previous examples to finitely presented groups of various finiteness types.

Contribution

It introduces a new family of finitely presented groups generalizing lamplighter groups, demonstrating quasi-isometric but not bilipschitz equivalence in higher rank cases.

Findings

Examples of finitely presented groups that are quasi-isometric but not bilipschitz.

Higher rank generalizations of lamplighter groups.

Groups of type F_n exhibiting these properties.

Abstract

We describe a family of finitely presented groups which are quasi-isometric but not bilipschitz equivalent. The first such examples were described by the first author and are the lamplighter groups $F \wr \mathbb{Z}$ where $F$ is a finite group; these groups are finitely generated but not finitely presented. The examples presented in this paper are higher rank generalizations of these lamplighter groups and include groups that are of type $F_n$ for any $n$.