Sofic boundaries and a-T-menability

TL;DR

This paper explores the relationship between the approximation properties of coarse boundary groupoids and the embeddability of graph sequences into Hilbert spaces, providing new characterizations of a-T-menability in both topological and measure-theoretic contexts.

Contribution

It establishes the equivalence between asymptotic coarse embeddability and topological a-T-menability of boundary groupoids, and relates measure-theoretic amenability to hyperfiniteness and property almost-A.

Findings

Asymptotic coarse embeddability is equivalent to topological a-T-menability.

Measure-theoretic amenability relates to hyperfiniteness and almost-A.

Results apply to spaces of graphs from sofic approximations.

Abstract

We undertake a systematic study of the approximation properties of the topological and measurable versions of the coarse boundary groupoid associated to a sequence of finite graphs of bounded degree. On the topological side, we prove that asymptotic coarse embeddability of the graph sequence into a Hilbert space is equivalent to the coarse boundary groupoid being topologically a-T-menable, thus answering a question by Rufus Willett. On the measure-theoretic side, we prove that measure-theoretic amenability resp. a-T-menability of the coarse boundary groupoid are related to hyperfiniteness and property almost-A resp. a version of "almost asymptotic embeddability into Hilbert space". These results can be directly applied to spaces of graphs coming from sofic approximations.