Edge sliding and ergodic hyperfinite decomposition

Abstract

We use edge slidings and saturated disjoint Borel families to give a conceptually simple proof of Hjorth's theorem on cost attained: if a countable p.m.p. ergodic equivalence relation $E$ is treeable and has cost $n \in \mathbb{N} \cup \{\infty\}$ then it is induced by an a.e. free p.m.p. action of the free group $\mathbb{F}_n$ on $n$ generators. More importantly, our techniques give a significant strengthening of this theorem: the action of $\mathbb{F}_n$ can be arranged so that each of the $n$ generators alone acts ergodically. The existence of an ergodic action for the first generator immediately follows from a powerful theorem of Tucker-Drob, whose proof however uses a recent substantial result in probability theory as a black box. We give a constructive and purely descriptive set theoretic proof of a weaker version of Tucker-Drob's theorem, which is enough for many of its applications, including our strengthening of Hjorth's theorem. Our proof uses new tools, such as asymptotic means on graphs, packed disjoint Borel families, and a cost threshold for finitizing the connected components of nonhyperfinite graphs.