Infinite graph product of groups II: Analytic properties

TL;DR

This paper investigates the analytic properties of graph products of finite groups with hyperbolic defining graphs, focusing on dynamics, convergence groups, and properties related to group von Neumann algebras.

Contribution

It introduces new classes of convergence groups, characterizes geometric finiteness conditions, and identifies properly proximal and bi-exact groups in this context.

Findings

Established a new class of convergence groups with hyperbolic graph products.

Provided necessary and sufficient conditions for geometric finiteness of convergence actions.

Identified new classes of properly proximal and bi-exact groups associated with graph products.

Abstract

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In particular, we provide a new class of convergence groups and identify the if and only if condition for this convergence action to be geometrically finite. We also provide a new class of properly proximal groups, relatively bi-exact groups, and groups with strongly solid group von Neumann algebras.