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 of de la Harpe.
Tullia Dymarz. "Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups." Duke Math. J. 154 (3) 509 - 526, 15 September 2010. https://doi.org/10.1215/00127094-2010-044