$\mathrm{L}^1$ full groups of flows

TL;DR

This work develops a comprehensive theory of L1 full groups for measure-preserving actions of Polish normed groups, unifying actions of discrete and locally compact groups under a single Polish-topology framework built from the L1-norm ||T||1 = ∫ D(x, Tx) dμ(x). It shows that topological derived subgroups are simple under mild hypotheses, and that for locally compact amenable acting groups they are whirly amenable and generically 2-generated; in the flow case, the index map identifies the derived subgroup as its kernel, yielding a precise abelianization and finite generation criteria tied to ergodic components. The paper also proves that L1 full groups remember L1 orbit equivalence classes, and establishes density and maximal-norm properties for the derived subgroups, along with detailed coarse-geometry results. A broad toolkit is developed, including cross-sections, Voronoi tessellations, Hopf decomposition, intermitted transformations, and coherent modifications, enabling a deep understanding of orbit structure, saturation phenomena, and orbitwise ergodicity in flows. Altogether, the results illuminate how L1 full groups encode orbit structure, ergodic decomposition, and coarse geometric features of measure-preserving flows and locally compact amenable actions, with implications for orbit equivalence and generation properties.

Abstract

We introduce the concept of an $\mathrm{L}^{1}$ full group associated with a measure-preserving action of a Polish normed group on a standard probability space. These groups carry a natural Polish group topology induced by an $\mathrm{L}^1$ norm. Our construction generalizes $\mathrm{L}^{1}$ full groups of actions of discrete groups, which have been studied recently by the first author. We show that under minor assumptions on the actions, topological derived subgroups of $\mathrm{L}^{1}$ full groups are topologically simple and -- when the acting group is locally compact and amenable -- are whirly amenable and generically two-generated. $\mathrm{L}^{1}$ full groups of actions of compactly generated locally compact Polish groups are shown to remember the $\mathrm{L}^{1}$ orbit equivalence class of the action. For measure-preserving actions of the real line (also often called measure-preserving flows), the topological derived subgroup of an $\mathrm{L}^{1}$ full groups is shown to coincide with the kernel of the index map, which implies that $\mathrm{L}^{1}$ full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. We also prove a reconstruction-type result: the $\mathrm{L}^{1}$ full group completely characterizes the associated ergodic flow up to flip Kakutani equivalence. Finally, we study the coarse geometry of the $\mathrm{L}^{1}$ full groups. The $\mathrm{L}^{1}$ norm on the derived subgroup of the $\mathrm{L}^{1}$ full group of an aperiodic action of a locally compact amenable group is proved to be maximal in the sense of C. Rosendal. For measure-preserving flows, this holds for the $\mathrm{L}^{1}$ norm on all of the $\mathrm{L}^{1}$ full group.

Figures (12)

• Figure 3.1: The involution $U_{n}$ is a products of $3$-cycles via $(12)(34) = (123)(234)$.
• Figure 7.1: An example of cross-sections $\mathcal{C}_{0}$ (all points), $\mathcal{C}_{1}$ (dots of size and above), $\mathcal{C}_{2}$ (marked as ) and $\mathcal{D}_{1}$, $\mathcal{D}_{2}$.
• Figure 7.2: Inductive step.
• Figure 7.3: Construction of the transformation $S$.
• Figure 8.1: Dynamics of a conservative orbit.

Theorems & Definitions (314)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4