Clinton Conley: Borel marker sets and hyperfiniteness

[this talk is a warmup for the Appalachian set theory workshop at CMU on October 24,
where Su Gao (UNT) will speak on “Countable abelian group actions”]

Mathematical logic seminar – September 29, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Borel marker sets and hyperfiniteness

Abstract:     A classical tool in ergodic theory is the Rokhlin lemma, which more or less states that any ergodic measure-preserving automorphism of a standard probability space is the uniform limit of periodic automorphisms. At its combinatorial core, the lemma’s proof relies on the ability to find measurable sets which intersect every orbit in a reasonably spaced out fashion. We discuss analogs of this in the purely Borel context, and use such marker sets to prove the Slaman-Steel / Sullivan-Weiss-Wright result that every Borel action of the integers on a standard Borel space generates a hyperfinite orbit equivalence relation. Time permitting, we discuss the (still open) problem of extending this to actions of arbitrary countable amenable groups, in preparation for Su Gao’s Appalachian Set Theory workshop this October.

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