Measure equivalence and sofic approximations

TL;DR

This paper develops a new technique to establish measure couplings between sofic groups, demonstrating that certain groups are exponentially measure equivalent despite not being quasi-isometric.

Contribution

Introduces a method for constructing measure couplings from sofic approximations, showing new measure equivalences among specific groups.

Findings

Solvable Baumslag-Solitar groups are exponentially measure equivalent to Lamplighter groups and SOL.

These groups are L^p measure equivalent for all p.

The groups are not quasi-isometric, highlighting differences between measure and geometric equivalence.

Abstract

We introduce a technique for producing a measure coupling between two sofic groups from a family of maps between their sofic approximations. We exploit this to construct measure couplings between pairs of groups with prescribed integrability conditions. As an application we show that solvable Baumslag-Solitar groups, Lamplighters and the group SOL are all exponentially measure equivalent to one another: in particular they are L^p measure equivalent for all p. This is in sharp contrast with the fact that these groups are in general not quasi-isometric to one another: indeed, for instance the lamplighter with lamp group Z/3Z is not quasi-isometric the lamplighter with lamp group Z/2Z.