On a measurable analogue of small topological full groups II

We pursue the study of ${\mathrm{L}}^1$ full groups of graphings and of the closures of their derived groups, which we call derived ${\mathrm{L}}^1$ 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 ${\mathrm{L}}^1$ full group has finite topological rank. We further show that a graphing is amenable if and only if its ${\mathrm{L}}^1$ full group is, and explain why various examples of (derived) ${\mathrm{L}}^1$ full groups fit very well into Rosendal’s geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between $\mathrm L^1$ full groups of amenable ergodic graphings must be a quasi-isometry for their respective ${\mathrm{L}}^1$ metrics. We finally show that ${\mathrm{L}}^1$ full groups of rank one transformations have topological rank 2.