Logic Seminar

Using the work of Hutchcroft and Nachmias on indistinguishability of the Wired Uniform Spanning Forest, R. Tucker-Drob recently proved a powerful theorem: any probability measure preserving locally countable ergodic Borel graph admits an ergodic hyperfinite subgraph. We give a completely different and self-contained proof of this theorem, which also goes through for a quasi-invariant probability measure, yielding a generalization of Tucker-Drob's theorem: any locally countable ergodic Borel graph on a standard probability space admits an ergodic hyperfinite subgraph. Our proof uses new tools whose combination yields an alternative way of exploiting nonamenability. In the first talk, we will state the main result and discuss one or two gadgets involved in the proof. In the second talk, we will finish the discussion of the new tools and give a sketch of our proof.