Baire measurable paradoxical decompositions via matchings

TL;DR

This paper proves that certain bipartite graphs and group actions with paradoxical decompositions can be realized with Baire measurable pieces, strengthening previous results and providing new solutions to the dynamical von Neumann-Day problem.

Contribution

It establishes Baire measurable paradoxical decompositions for group actions and bipartite graphs, extending prior work and solving related problems in descriptive set theory and dynamical systems.

Findings

Existence of Borel perfect matchings under strengthened Hall's condition.

Paradoxical decompositions can be realized with Baire measurable pieces.

A Baire category solution to the dynamical von Neumann-Day problem.

Abstract

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann-Day problem: if $a$ is a nonamenable action of a group on a Polish space $X$ by Borel automorphisms, then there is a free Baire measurable action of $\mathbb{F}_2$ on $X$ which is Lipschitz with respect to $a$.