Dynamical properties of profinite actions

TL;DR

This paper investigates the properties of profinite actions of residually finite groups, establishing conditions for weak equivalence and constructing many inequivalent free actions, while also exploring property (τ) inheritance and implications for expander graphs.

Contribution

It proves that strongly ergodic profinite actions are classified by isomorphism, constructs many weakly inequivalent free actions, and analyzes property (τ) inheritance and graph theoretical consequences.

Findings

Strongly ergodic profinite actions are weakly equivalent iff they are isomorphic.

Constructs continuum many pairwise weakly inequivalent free actions.

Shows that expander covering towers are either bipartite or far from bipartite.

Abstract

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($\tau$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($\tau$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.