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.