Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups

TL;DR

This paper introduces a new product theorem for sublinear bilipschitz equivalences, enabling the classification of solvable Lie groups up to quasiisometry and distinguishing their geometric structures more precisely.

Contribution

It generalizes classical quasiisometry results to sublinear bilipschitz equivalences and applies this to classify certain solvable Lie groups beyond previous methods.

Findings

Established a product theorem for sublinear bilipschitz equivalences.

Differentiated families of solvable groups up to quasiisometry using the new theorem.

Confirmed the existence of uncountably many quasiisometry classes of certain 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 R\'emy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.