Scott Schneider: Commuting endomorphisms and hypersmooth equivalence relations

Thursday, September 22, 4:00-5:30, East Hall 3096.

An equivalence relation E is hypersmooth (hyperfinite) if E is the union of an increasing sequence of smooth (finite) Borel equivalence relations.  In the mid 80s, Weiss proved that the equivalence relation generated by a finite family of commuting Borel automorphisms is hyperfinite, and in the mid 90s, Dougherty, Jackson, and Kechris proved that the equivalence relation generated by a single Borel endomorphism is hypersmooth.  We will generalize both results to show that the equivalence relation generated by a finite family of commuting Borel endomorphisms is hypersmooth.  As is typical in this area, the proof will involve the construction of a suitable family of Borel marker sets.  This talk will be part 1 of 2.