A generalization of the Davis-Moussong complex for Dyer groups

TL;DR

This paper introduces Dyer groups, a broad class including Coxeter and right-angled Artin groups, and constructs a CAT(0) complex generalizing known complexes, providing new insights into their geometric structure.

Contribution

The paper defines Dyer groups, shows they are finite index subgroups of Coxeter groups, and constructs a generalized CAT(0) complex extending Davis-Moussong and Salvetti complexes.

Findings

Dyer groups include Coxeter and right-angled Artin groups

Constructed a generalized CAT(0) complex for Dyer groups

Proved the complex is CAT(0)

Abstract

A common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. Matthew Dyer introduced a class of groups, which we call Dyer groups, sharing this feature. This class includes, but is not limited to, Coxeter groups, right-angled Artin groups, and graph products of cyclic groups. We introduce Dyer groups by giving their standard presentation and show that they are finite index subgroups of Coxeter groups. We then introduce a piecewise Euclidean cell complex $\Sigma$ which generalizes the Davis-Moussong complex and the Salvetti complex. The construction of $\Sigma$ uses simple categories without loops and complexes of groups. We conclude by proving that the cell complex $\Sigma$ is CAT(0).