Proper proximality for various families of groups

TL;DR

This paper studies and classifies proper proximality in various group families, linking geometric actions, wreath products, and graph products, and relates these to rigidity and von Neumann algebra properties.

Contribution

It provides new classifications of proper proximality for groups acting on trees, wreath products, and graph products, extending previous work on inner amenability.

Findings

Groups acting on trees with finite edge stabilizer intersections are properly proximal.

Wreath product G wr H is properly proximal iff H is non-amenable.

Complete classification of proper proximality among graph products.

Abstract

In this paper, the notion of proper proximality (introduced in [BIP18]) 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\wr 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 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.