EVENT DETAILS AND ABSTRACT

Title: Ergodic hyperfinite decomposition of pmp equivalence relations
Speaker: Anush Tserunyan
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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 \emph{graphing} of $E$. When $E$ is probability measure preserving (pmp), graphings provide a numerical invariant called \emph{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 pmp, ergodic, treeable (admits an acyclic graphing), and has cost $n \in \mathbb{N} \cup \{\infty\}$, then it is generated by an a.e. free measure-preserving action of the free group $\mathbf{F}_n$ on $n$ generators. Jointly with Benjamin Miller, we develope new techniques of modifying the graphing, which yield a strengthening of this theorem: the action of $\mathbf{F}_n$ can be arranged so that each of the $n$ generators alone acts ergodically.