Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups

TL;DR

This paper demonstrates that quasi-isometric groups can fail to be bilipschitz equivalent, providing a counterexample in geometric group theory and answering a previously open question.

Contribution

It establishes that bilipschitz equivalence is strictly stronger than quasi-isometry for certain finitely generated groups, specifically lamplighter groups.

Findings

Quasi-isometric lamplighter groups are not bilipschitz equivalent.

Provides a positive answer to an open question in geometric group theory.

Highlights differences between bilipschitz and quasi-isometric classifications.

Abstract

We show that certain lamplighter groups that are quasi-isometric to each other are not bilipschitz equivalent. This gives a positive answer to a question in Topics in Geometric Group Theory by Pierre de la Harpe (page 107).