Bilipschitz versus quasi-isometric equivalence for higher rank lamplighter groups

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 (Duke Math. J., 2010) and are the lamplighter groups F integral 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.