Seminar organizers: William Chong, Katherine Goldman, Christopher Karpinski, Carl Kristof-Tessier, Piotr Przytycki, Daniel Wise
A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Recently, a lot of research has been done on hyperfiniteness of the orbit equivalence relation on the Gromov boundary induced by various group actions on hyperbolic spaces. In this talk, I will explain my attempt to explore this connection of Borel complexity and geometric group theory for another intensively studied geometric object, which is CAT(0) cube complexes. More precisely, we prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite.
I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space $\text{BDiff}(M)$, for M a compact, connected, reducible 3-manifold. We prove that when $M$ is orientable and has non-empty boundary, $\text{BDiff}(M\text{ rel } ∂M)$ has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where $M$ is irreducible by Hatcher and McCullough. The theory we develop to prove this theorem has other applications, and I'll provide an overview of these.