Baire measurable paradoxical decompositions via matchings

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 $a$.

Minor revisions

References in corpus (2)

• Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces
• Measurable equidecompositions via combinatorics and group theory

Cited by in corpus (7)

• Measurable equidecompositions for group actions with an expansion property
• On Følner sets in topological groups
• About von Neumann's problem for locally compact groups
• Measurable perfect matchings for acyclic locally countable Borel graphs
• On Baire Measurable Colorings of Group Actions
• Measurable realizations of abstract systems of congruences
• Hyperfiniteness on Topological Ramsey Spaces