流的$\text{L}^1$全群
$\mathrm{L}^1$ full groups of flows
AI总结：
本文引入波兰赋范群保测作用的$\text{L}^1$全群概念，推广离散群相关构造，证明其拓扑导子群的拓扑单性、可 amenable 性等性质，刻画流的遍历分支与生成性关联，还研究粗几何及范数极大性。
摘要：
我们引入了与波兰赋范群在标准概率空间上的保测作用相关的$\text{L}^1$全群概念。这些群带有由$\text{L}^1$范数诱导的自然波兰群拓扑。我们的构造推广了离散群作用的$\text{L}^1$全群，后者近期已由第一作者研究。我们证明，在对作用的次要假设下，$\text{L}^1$全群的拓扑导子群是拓扑单群；当作用群是局部紧且可 amenable 时，它们是旋转可 amenable 且一般由两个元生成。紧生成局部紧波兰群作用的$\text{L}^1$全群被证明能记住作用的$\text{L}^1$轨道等价类。对于实直线的保测作用（也常称为保测流），$\text{L}^1$全群的拓扑导子群被证明与指标映射的核一致，这意味着自由保测流的$\text{L}^1$全群是拓扑有限生成的当且仅当该流允许有限多个遍历分支。我们还证明了一个重构型结果：$\text{L}^1$全群完全刻画了相关遍历流的翻转 Kakutani 等价类。最后，我们研究了$\text{L}^1$全群的粗几何。局部紧 amenable 群的非周期作用的$\text{L}^1$全群的导子群上的$\text{L}^1$范数被证明在 C. Rosendal 意义下是极大的；对于保测流，这对整个$\text{L}^1$全群上的$\text{L}^1$范数成立。
英文摘要：
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.
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