$\mathrm{L}^1$ full groups of flows
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917986399420416 |
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| author | Maître, François Le Slutsky, Konstantin |
| author_facet | Maître, François Le Slutsky, Konstantin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_09009 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $\mathrm{L}^1$ full groups of flows Maître, François Le Slutsky, Konstantin Dynamical Systems Functional Analysis Group Theory 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. |
| title | $\mathrm{L}^1$ full groups of flows |
| topic | Dynamical Systems Functional Analysis Group Theory |
| url | https://arxiv.org/abs/2108.09009 |