Sofic approximations and quantitative measure couplings

TL;DR

This paper explores the inverse problem of quantitative measure subgroup couplings within the framework of measure equivalence, focusing on the lamplighter group to understand geometric refinements of this relation.

Contribution

It provides an analysis of the inverse problem for quantitative measure subgroup couplings specifically for the lamplighter group, advancing the understanding of geometric refinements in measure equivalence.

Findings

Solved the inverse problem for the lamplighter group.

Established conditions for measure subgroup couplings with prescribed groups.

Enhanced understanding of quantitative measure equivalence in geometric group theory.

Abstract

Measure equivalence was introduced by Gromov as a measured analogue of quasi-isometry. Unlike the latter, measure equivalence does not preserve the large scale geometry of groups and happens to be very flexible in the amenable world. Indeed the Ornstein-Weiss theorem shows that all infinite countable amenable groups are measure equivalent to the group of integers. To refine this equivalence relation and make it responsive to geometry, Delabie, Koivisto, Le Ma\^itre and Tessera introduced a quantitative version of measure equivalence. They also defined a relaxed version of this notion called quantitative measure subgroup coupling. In this article we offer to answer the inverse problem of the quantification (find a group admitting a measure subgroup coupling with a prescribed group with prescribed quantification) in the case of the lamplighter group.