Set theory workshop
Anush Tserunyan
UIUC
Integer cost and ergodic actions
Abstract: A countable Borel equivalence relation $E$ on a probability space can always be generated in two ways: as the orbit equivalence relation of a Borel action of a countable group and as the connectedness relation of a locally countable Borel graph, called a $\it{graphing}$ of $E$. Assuming that $E$ is measure-preserving, graphings provide a numerical invariant called $\it{cost}$, whose theory has been largely developed and used by Gaboriau and others in establishing rigidity results. A well-known theorem of Hjorth states that when $E$ is ergodic, treeable (admits an acyclic graphing), and has integer or infinite cost $n \le \infty$, then it is generated by an a.e. free measure-preserving action of the free group $\mathbf{F}_n$ on $n$ generators. We give a simpler proof of this theorem and the technique of our proof, combined with two other new tools, yields a strengthening of Hjorth's theorem: the action of $\mathbf{F}_n$ can be arranged so that each of the $n$ generators acts ergodically. This is joint work with Benjamin Miller.
Saturday October 22, 2016 at 11:00 AM in SEO 636