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

Koichi Oyakawa (Vanderbilt)

Abstract: In this talk, I will introduce the extension graph of graph product of groups and explain its geometry. This notion enables us to study the properties of graph product by exploiting the large-scale geometry of its defining graph. In particular, I show that the asymptotic dimension of the extension graph exhibits the same behavior as in the case of quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. In addition, I present applications of the extension graph to the study of convergence actions, graph wreath products, and group von Neumann algebras when the defining graph is hyperbolic.