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- Title
Proper proximality among various families of groups.
- Authors
Changying Ding; Elayavalli, Srivatsav Kunnawalkam
- Abstract
In this paper, the notion of proper proximality (introduced by Boutonnet, Ioana, and Peterson [Ann. Sci. Éc. Norm. Supér. (4) 54 (2021), 445-482]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that, for any two edges, the intersection of their edge stabilizers is finite, then G is properly proximal. We show that the wreath product G ... H is properly proximal if and only if H is non-amenable. We then completely classify proper proximality among graph products of non-trivial groups. Our results generalize the recent work of Duchesne, Tucker-Drob, and Wesolek classifying inner amenability for these families of groups. Our results also recover some rigidity results associated to the group von Neumann algebras by virtue of being properly proximal. A key idea in the proofs of our theorems is a technique to upgrade from relative proper proximality using computations in the double dual of the small at infinity boundary.
- Subjects
VON Neumann algebras; TREES
- Publication
Groups, Geometry & Dynamics, 2024, Vol 18, Issue 3, p921
- ISSN
1661-7207
- Publication type
Article
- DOI
10.4171/GGD/778