Geometric property (T) for box spaces and sofic approximations

TL;DR

This paper establishes connections between geometric property (T), sofic approximations, and box spaces, showing how these properties relate to spectral gaps and providing criteria for geometric property (T).

Contribution

It demonstrates that sofic approximations of property (T) groups can be approximated by groups with geometric property (T), and introduces a local geometric criterion for this property.

Findings

Sofic approximations of property (T) groups are approximately isomorphic to groups with geometric property (T).

A sequence of bounded degree graphs is approximately isomorphic to a union of expanders iff the Laplacian has a spectral gap.

A local geometric criterion for geometric property (T) is established, akin to uk's criterion.

Abstract

We prove that every sofic approximation of a property (T) group is approximately isomorphic to one having geometric property (T), and more generally, a box space of graphs which has boundary geometric property (T) is approximately isomorphic to one having geometric property (T). We also prove that a sequence of bounded degree graphs is approximately isomorphic to a disjoint union of expanders if and only if the Laplacian has spectral gap in the ultraproduct. Finally, we prove a local geometric criterion for geometric property (T) in the spirit of \.{Z}uk's criterion for property (T) for groups.