On number of ends of graph products of groups

TL;DR

This paper characterizes the number of ends of graph products of finitely generated groups based on the combinatorial properties of the underlying graph, providing a complete classification.

Contribution

It offers a complete characterization of the number of ends of graph products of finitely generated groups, linking algebraic properties to graph combinatorics.

Findings

Complete classification of ends for graph products

Conditions on the graph determine the number of ends

Links between graph structure and group ends

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.