RAAGedy right-angled Coxeter groups

TL;DR

This paper provides criteria and methods to determine when a triangle-free graph defines a right-angled Coxeter group that is quasiisometric to a right-angled Artin group, introducing new graph operations and computational tools.

Contribution

It introduces new graph modification operations and develops criteria for identifying quasiisometric relationships between Coxeter and Artin groups.

Findings

Criteria for identifying such graphs are established.

New graph operations, cloning and unfolding, are introduced.

The methods are implemented and successfully applied to small graphs.

Abstract

We give criteria for deciding whether or not a triangle-free simple graph is the presentation graph of a right-angled Coxeter group that is quasiisometric to some right-angled Artin group, and, if so, producing a presentation graph for such a right-angled Artin group. We introduce two new graph modification operations, cloning and unfolding, to go along with an existing operation called link doubling. These operations change the presentation graph but not the quasiisometry type of the resulting group. We give criteria on the graph that imply it can be transformed by these operations into a graph that is recognizable as presenting a right-angled Coxeter group commensurable to a right-angled Artin group. In the converse direction we derive coarse geometric obstructions to being quasiisometric to a right-angled Artin group, first by specializing existing results from the literature to this setting, then by developing new approaches using configurations of maximal product regions. In all cases we give sufficient graphical conditions that imply these geometric obstructions. We implemented our criteria on a computer and applied them to an enumeration of small graphs. Our methods completely answer the motiving question when the graph has at most 10 vertices.