Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups

Abstract

We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. We employ our product theorem to distinguish up to quasiisometry certain families of solvable groups which share the same dimension, cone-dimension and Dehn function; actually we do this by distinguishing them up to sublinear bilipschitz equivalence, which is slightly stronger. As an application, we recover the fact, recently obtained by Bourdon and Rémy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.

v1->v2: v1 has been split in two parts. v2 contains the first part, after revision and the addition of Corollary D. The second part of v1 can be found at arXiv:2509.12823. v2->v3: Revised following a referee report, corrections in Section 2 and weakening of the claim about the error term in the conclusion of Theorem A