# Event Detail

 Fri Feb 20, 2015 60 Evans Hall, 4–6 PM Logic Colloquium Andrew Marks (California Institute of Technology) Baire measurable paradoxical decompositions and a Baire category solution to the dynamical von Neumann-Day problem

The Banach-Tarski paradox states that the unit ball in \$\mathbb{R}^3\$ is equidecomposable with two unit balls in \$\mathbb{R}^3\$ by rigid motions. In 1930, Marczewski asked whether there is such an equidecomposition where each piece has the Baire property. Using an intricate construction, Dougherty and Foreman gave a positive answer to this question.

We generalize Dougherty and Foreman’s result to completely characterize which Borel group actions have Baire measurable paradoxical decompositions. We 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. We also obtain a Baire category solution to the dynamical von Neumann-Day problem, in the spirit of Whyte and Gaboriau-Lyons: 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\$. The main tool we use to prove these theorems is a version of Hall’s matching theorem for Borel graphs.

This is joint work with Spencer Unger at UCLA.