The number of topological generators for full groups of ergodic equivalence relations

TL;DR

This paper explores the relationship between the cost and the minimal number of topological generators of full groups in ergodic pmp equivalence relations, providing exact results for free group actions and answering a key open question.

Contribution

It establishes a complete connection between cost and topological generators for ergodic equivalence relations, including a precise answer for free group actions.

Findings

The minimal number of topological generators for the full group of a free pmp ergodic action of a free group on n generators is n+1.

The paper answers a question posed by Kechris regarding these invariants.

It clarifies the relationship between cost and topological generators in the context of ergodic equivalence relations.

Abstract

We completely elucidate the relationship between two invariants associated with an ergodic probability measure-preserving (pmp) equivalence relation, namely its cost and the minimal number of topological generators of its full group. It follows that for any free pmp ergodic action of the free group on $n$ generators, the minimal number of topological generators for the full group of the action is $n+1$, answering a question of Kechris.