A generalization of the Davis–Moussong complex for Dyer groups

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 $\mathrm{CAT}(0)$.