Geometric property (T) for box spaces and sofic approximations

TL;DR

The paper develops a framework linking coarse geometric properties of box spaces to analytic properties of groups via coarse boundary groupoids and ultraproducts. It proves that almost boundary geometric property $T$ implies approximate isomorphism to a box space with geometric $T$, and establishes equivalent conditions for sofic groups connecting $T$, almost boundary $T$, and geometric $T$ approximations. A spectral-gap criterion in ultraproducts characterizes when a box space is approximately an expander, enabling a decomposition of geometric $T$ into expander behavior and boundary $T$, and it provides a local Żuk-type criterion for box spaces guaranteeing geometric $T$. The results yield a robust connection between sofic approximations and expanders, with consequences for measured versus topological $T$, and open questions about randomized constructions and sums-of-squares characterizations of geometric $T$.

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 Żuk's criterion for property (T) for groups.

Theorems & Definitions (26)

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• proof : Proof of Proposition \ref{['exp+bdr']}
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