On number of ends of graph products of groups

Abstract

Given a finite simplicial graph $\Gamma=(V,E)$ with a vertex-labelling $\varphi:V\rightarrow\left\{\text{non-trivial finitely generated groups}\right\}$, the graph product $G_\Gamma$ is the free product of the vertex groups $\varphi(v)$ with added relations that imply elements of adjacent vertex groups commute. For a quasi-isometric invariant $\mathcal{P}$, we are interested in understanding under which combinatorial conditions on the graph $\Gamma$ the graph product $G_\Gamma$ has property $\mathcal{P}$. In this article our emphasis is on number of ends of a graph product $G_\Gamma$. In particular, we obtain a complete characterization of number of ends of a graph product of finitely generated groups.