Sofic boundaries of groups and coarse geometry of sofic approximations

TL;DR

This paper introduces a topological notion of sofic boundaries for groups, linking coarse geometric properties of sofic approximations to analytic properties of the groups themselves.

Contribution

It generalizes existing results from residually finite groups to sofic groups, establishing new connections between coarse geometry and group properties.

Findings

Coarse properties of sofic approximations imply group properties like amenability and property (T).

Coarse equivalence of sofic approximations leads to measure equivalence between groups.

Provides a geometric perspective on ultralimits of finite graphs and measure structures on groupoids.

Abstract

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric view point on ultralimits of a sequence of finite graphs first exposed by Jan \v{S}pakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists.