论文标题
从翘曲锥的粗糙几何形状到测量的组耦合
From the coarse geometry of warped cones to the measured coupling of groups
论文作者
论文摘要
在本文中，我们证明，如果两个弯曲的锥体对应于两个有限，等距，测量的有限产生的组，则在两个带有概率措施的紧凑型公制空间上进行的操作将是级别的准静电（有一些额外的自然假设），那么相应的组是均匀测量的（UME）。从de laat-vigolo和Sawicki的作品中得知，如果两个这样的翘曲锥体是级别的准等级，那么它们的稳定产品将是准时的。我们加强了这一结果，并进一步证明了团体的UME。我们还讨论了我们主要结果的许多应用。我们给出了群体和相关的翘曲锥的无数实例，使组是互相矛盾的，但是从我们的主要定理的意义上讲，翘曲的锥并不是相互准时的。我们还提供了两个弯曲锥的示例（对于两个不同的扩展器家族，它们是准时的），因此，从我们的主要定理的意义上讲，它们中的一个不会准确地嵌入另一个。
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.
