2017/04/20 by Benjamin D. Miller, Miller, Benjamin D., Anush Tserunyan +1 · 1 citation
Mathematics · #03E15 #05C63 #28D05 #37A05 #37A20 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.1704.06019
openalex publication_date 2017/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∈ ℕ ∪ \∞\ then it is induced by an a.e. free p.m.p. action of the free group \mathbbFn on n generators. More importantly, our techniques give a significant strengthening of this theorem: the action of \mathbbFn 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.