Abstract:
Given a countable group G acting measurably on a space X, one can consider the equivalence relation on X induced by orbits.
When the space X is endowed with a measure, and G is amenable, the fundamental works of Ornstein-Weiss, and Connes-Feldman-Weiss showed that this equivalence relation is "hyperfinite" up to a set of measure zero, meaning it can be measurably exhausted by finite chunks of orbits in some sense.
Together with Dye's theorem, this shows that all ergodic, p.m.p. group actions of amenable groups are orbit equivalent, a celebrated powerful result.
Subsequently, Benjy Weiss asked whether orbit equivalence relations of amenable group actions should be hyperfinite, but without discarding a set of measure zero, now known as Weiss' question.
We will introduce some of the ideas of the paper "Borel Asymptotic Dimension and Hyperfinite Equivalence Relations" of C. Conley, S. Jackson, A. Marks, B. Seward, R. Tucker-Drob, which has significantly advanced the state of this question, and connected it more precisely to the study of coarse geometry of groups. Time Permitting, we will give a short proof of the fact that all Borel actions of nilpotent groups are hyperfinite.