Outer automorphism groups of equivalence relations for mapping class group actions

Abstract

We study outer automorphism groups Out(ℛ) of equivalence relations ℛ arising from ergodic, essentially free and measure-preserving actions on standard probability spaces of mapping class groups, and compute them explicitly for some examples. We show that the subgroups of Out(ℛ) associated with action-automorphisms are always finite index ones. Moreover, a sharp upper bound of the indices among all such actions of mapping class groups is obtained.