Geometry and dynamics of the extension graph of graph product of groups

Abstract

We introduce the extension graph of graph product of groups and study its geometry. This enables us to study properties of graph products of groups by exploiting large scale geometry of its defining graph. In particular, we show that the extension graph exhibits the same phenomenon about asymptotic dimension as quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. Moreover, we present three applications of the extension graph of graph product when a defining graph is hyperbolic. First, we provide a new class of convergence groups by considering the action of graph product of finite groups on a compactification of the extension graph and identify the if and only if condition for this action to be geometrically finite. Secondly, we prove relative hyperbolicity of the semi-direct product of groups that interpolates between wreath product and free product. Finally, we provide a new class of graph product of finite groups whose group von Neumnann algebra is strongly solid.