On a measurable analogue of small topological full groups II

TL;DR

This paper investigates the properties of $ ext{L}^1$ full groups associated with graphings, establishing connections between their algebraic structure, entropy, amenability, and geometric properties, with implications for isomorphisms and topological rank.

Contribution

It introduces the concept of derived $ ext{L}^1$ full groups, characterizes their properties in relation to entropy and amenability, and explores their geometric and topological structure.

Findings

Finite Rokhlin entropy corresponds to finite topological rank of derived $ ext{L}^1$ full groups.

A graphing is amenable if and only if its $ ext{L}^1$ full group is amenable.

$ ext{L}^1$ full groups of rank one transformations have topological rank 2.

Abstract

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.