Cost of inner amenable equivalence relations

Cost is a $[1,\infty]$-valued measure-isomorphism invariant of aperiodic measured equivalence relations defined by Gilbert Levitt and heavily studied by Damien Gaboriau. For a large class of equivalence relations, including amenable, the cost is 1. Yoshikata Kida and Robin Tucker-Drob recently defined the notion of an inner amenable equivalence relation as an analog of inner amenability in the setting of groups. I will give an overview of the theory of cost and inner amenable groups as well as several previous related results. If time permits, I'll outline a proof that inner amenable equivalence relations also have cost 1. This is joint work with Robin Tucker-Drob.