Logic Seminar

Please note that the time is PST

We study measure-class-preserving (mcp) equivalence relations and seek criteria for their (non)amenability. Such criteria are well established for probability-measure-preserving (pmp) equivalence relations, where tools like cost and ℓ^2-Betti numbers are available. However, in the mcp setting, these tools are absent and much less is known. We discuss a recently developed structure theory for mcp equivalence relations, including a precise characterization of amenability for treed mcp equivalence relations in terms of the interplay between the geometry of the trees and the Radon–Nikodym cocycle. This generalizes Adams' dichotomy to the mcp setting and yields a complete description of the structure of amenable subrelations of treed equivalence relations, as well as anti-treeability results. We also establish a Day–von Neumann-type result for multi-ended mcp graphs, generalizing a theorem of Gaboriau and Ghys. Joint work with Robin Tucker-Drob, and with Ruiyuan Chen and Grigory Terlov.