从 warped cones 的粗几何到群的测度耦合
From the coarse geometry of warped cones to the measured coupling of groups
AI总结：
本文研究 warped cones 的层级拟等距如何反映群的测度论关系，证明在自然条件下这蕴含相应有限生成群一致测度等价，并给出区分 Warped cones 与 expander 族嵌入性的例子。
摘要：
在本文中，我们证明：如果对应于两个有限生成群的两个 warped cones 在各层级上拟等距（并带有一些额外的自然假设），其中这两个群在两个带概率测度的紧致度量空间上具有自由、等距且保测度的作用，那么相应的群是一致测度等价（UME）的。早先从 de Laat–Vigolo 和 Sawicki 的工作中已经知道，如果两个这样的 warped cones 在各层级上拟等距，那么它们的稳定积是拟等距的。我们加强了这一结果，并进一步证明了这些群的 UME。我们还讨论了主要结果的许多应用。我们给出了可数无穷多个群及相关 Warped cones 的例子，使得这些群彼此拟等距，但按照我们主要定理的意义，这些 Warped cones 彼此并不拟等距。我们还提供了两个 Warped cones（它们分别拟等距于两个不同的 expander 族）的例子，使得按照我们主要定理的意义，其中一个不能拟等距嵌入另一个。
英文摘要：
In this article, we prove that if two warped cones corresponding to two finitely generated groups with free, isometric, measure-preserving, actions on two compact metric spaces with probability measures are level-wise quasi-isometric (with some extra natural assumptions), then the corresponding groups are uniformly measured equivalent (UME). It was earlier known from the works of de Laat-Vigolo and Sawicki that if two such warped cones are level-wise quasi-isometric, then their stable products are quasi-isometric. We strengthen this result and go further to prove UME of the groups. We also discuss many applications of our main result. We give countably infinite examples of groups and associated Warped cones such that the groups are mutually quasi-isometric, but the Warped cones are not mutually quasi-isometric in the sense of our main theorem. We also provide examples of two Warped cones (which are quasi-isometric to two different expander families) such that one of them does not quasi-isometrically embed into the other one in the sense of our main theorem.
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