Proper proximality among various families of groups (Q6594553)

Informally, a group \(G\) is said to be properly proximal if it admits finitely many non-trivial continuous actions of \(G\) on compact spaces \(K_{i}\) such that any sequence in \(G\) admits a subsequence which is in some sense proximal for at least one of the actions of \(G\) on \(K_{i}\). This concept was introduced by \textit{R. Boutonnet} et al. in [Ann. Sci. Éc. Norm. Supér. (4) 54, No. 2, 445--482 (2021; Zbl 07360850)], where the precise definition can be found. The class of properly proximal groups contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups.\N\NIn the paper under review the authors show that if a group \(G\) 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. They prove that the wreath product \(G \wr H\) is properly proximal if and only if \(H\) is non-amenable. Furthermore, they classify completely classify proper proximality among graph products of non-trivial groups. Such results also recover some rigidity results associated with the group von Neumann algebras by virtue of being properly proximal.