Infinite graph product of groups I: Geometry of the extension graph

TL;DR

This paper introduces the extension graph for graph products of groups, revealing its geometric properties and applications, including the relative hyperbolicity of certain complex group constructions, thereby advancing understanding of large-scale geometry in group theory.

Contribution

It defines the extension graph for graph products, shows its isomorphism to the crossing graph of a quasi-median graph, and applies this to prove relative hyperbolicity of graph-wreath products.

Findings

Extension graph is isomorphic to the crossing graph of a quasi-median graph.

Extension graph exhibits asymptotic dimension phenomena similar to quasi-trees.

Proves relative hyperbolicity of graph-wreath products.

Abstract

We introduce the extension graph of graph product of groups and study its geometry. This enables us to study properties of graph product by exploiting large scale geometry of its defining graph. In particular, we show that the extension graph is isomorphic to the crossing graph of a canonical quasi-median graph and exhibits the same phenomenon about asymptotic dimension as quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. As an application of the extension graph, we prove relative hyperbolicity of graph-wreath product. This provides a new construction of relatively hyperbolic groups.