| dc.description.abstract | Orbit equivalence is an equivalence relation on measurable actions of groups that’s been studied since the 1950’s. It has connections to many areas of mathematics including descriptive set theory, percolation theory, ergodic theory, representation theory, von neumann algebras, and geometric group theory. In joint work with Robin Tucker-Drob, we show inner amenable groupoids have fixed price 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner amenable groups have fixed price 1. We later study coamenable inclusions of inner amenable groupoids to generalize a result from the setting of groups. We also prove several equivalent conditions to amenability of an action of a groupoid. In additional joint work with Robin Tucker-Drob, we study wreath products up to orbit equivalence and show that C2 ≀ F2 is orbit equivalent to Cn ≀ F2. In order to accomplish this, we introduce and study the notion of cofinitely equivariant maps. We also prove some examples of rigidity in this setting. | en |
