On a measurable analogue of small topological full groups II

Abstract

We pursue the study of L 1 full groups of graphings and of the closures of their derived groups, which we call derived L 1 full groups. Our main result shows that aperiodic probability measure-preserving actions of finitely generated groups have finite Rokhlin entropy if and only if their derived L 1 full group has finite topological rank. We further show that a graphing is amenable if and only if its L 1 full group is, and explain why various examples of (derived) L 1 full groups fit very well into Rosendal’s geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between L 1 full groups of amenable ergodic graphings must be a quasi-isometry for their respective L 1 metrics. We finally show that L 1 full groups of rank one transformations have topological rank 2.