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 R3 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 F2 on X which is Lipschitz with respect to a.