On full groups of non ergodic probability measure preserving equivalence relations

TL;DR

This paper explores the structure of full groups of non-ergodic probability measure preserving equivalence relations, relating their topological rank to ergodic components, and investigates properties like automatic continuity and extreme amenability.

Contribution

It introduces a formula linking topological rank to ergodic components, constructs examples with dense free subgroups, and characterizes aperiodicity via algebraic properties.

Findings

Topological rank relates to the cost of ergodic components.

Existence of dense free subgroups matching the topological rank.

Full groups of hyperfinite relations are extremely amenable.

Abstract

We give a formula relating the topological rank of the full group of an aperiodic pmp equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups having a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to give an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing to the non ergodic case a result of Giordano and Pestov.