Inapproximability of actions and Kazhdan's property (T)

TL;DR

This paper explores the limitations of approximating certain group actions with finite graphs, revealing new structural properties related to Kazhdan's property (T) and sofic approximations.

Contribution

It constructs specific group actions that cannot be approximated by finite graphs and introduces a novel approach using almost automorphisms of sofic approximations.

Findings

Constructed non-approximable group actions

Showed the set of epsilon-automorphisms forms a group

Linked properties of Kazhdan groups with sofic approximations

Abstract

We construct p.m.p. group actions that are not local-global limits of sequences of finite graphs. Moreover, they do not weakly contain any sequence of finite labeled graphs. Our methods are based on the study of almost automorphisms of sofic approximations: We show that the set of epsilon-automorphisms of a sufficiently good sofic approximation of a Kazhdan group by expanders form a group in a natural way.