Piecewise strongly proximal actions, free boundaries and the Neretin groups

TL;DR

This paper establishes a dynamical criterion for totally disconnected groups acting on compact spaces, showing that Neretin groups have specific representation properties and are not of type I.

Contribution

It introduces a new criterion linking group actions to the confinement of subgroups and applies it to Neretin groups, revealing their representation theory properties.

Findings

Neretin groups have two inequivalent irreducible unitary representations that are weakly equivalent.

Neretin groups are not of type I.

The criterion ensures no relatively amenable subgroup can be confined under certain actions.

Abstract

A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary representations that are weakly equivalent. This implies that the Neretin groups are not of type I, thereby answering a question of Y.~Neretin.