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.
Changying Ding; Srivatsav Kunnawalkam Elayavalli. Proper proximality among various families of groups. Groups, geometry, and dynamics, Tome 18 (2024) no. 3, pp. 921-938. doi: 10.4171/ggd/778
@article{10_4171_ggd_778,
author = {Changying Ding and Srivatsav Kunnawalkam Elayavalli},
title = {Proper proximality among various families of groups},
journal = {Groups, geometry, and dynamics},
pages = {921--938},
year = {2024},
volume = {18},
number = {3},
doi = {10.4171/ggd/778},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/778/}
}
TY - JOUR
AU - Changying Ding
AU - Srivatsav Kunnawalkam Elayavalli
TI - Proper proximality among various families of groups
JO - Groups, geometry, and dynamics
PY - 2024
SP - 921
EP - 938
VL - 18
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/778/
DO - 10.4171/ggd/778
ID - 10_4171_ggd_778
ER -
%0 Journal Article
%A Changying Ding
%A Srivatsav Kunnawalkam Elayavalli
%T Proper proximality among various families of groups
%J Groups, geometry, and dynamics
%D 2024
%P 921-938
%V 18
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/778/
%R 10.4171/ggd/778
%F 10_4171_ggd_778
