Word problem and parabolic subgroups in Dyer groups

TL;DR

This paper explores the word problem solutions and parabolic subgroup structures in Dyer groups, a family that generalizes Coxeter groups and right-angled Artin groups, revealing new subgroup intersection properties.

Contribution

It introduces quasi-Dyer groups, extends the solution to the word problem to these groups, and analyzes parabolic subgroup intersections in Dyer groups.

Findings

All Dyer groups have a solution to the word problem.

Groups with this solution are contained in the quasi-Dyer group family.

Intersections of parabolic subgroups in finite-type Dyer groups are parabolic.

Abstract

One can observe that Coxeter groups and right-angled Artin groups share the same solution to the word problem. On the other hand, in his study of reflection subgroups of Coxeter groups Dyer introduces a family of groups, which we call Dyer groups, which contains both, Coxeter groups and right-angled Artin groups. We show that all Dyer groups have this solution to the word problem, we show that a group which admits such a solution belongs to a little more general family of groups that we call quasi-Dyer groups, and we show that this inclusion is strict. Then we show several results on parabolic subgroups in quasi-Dyer groups and in Dyer groups. Notably, we prove that any intersection of parabolic subgroups in a Dyer group of finite type is a parabolic subgroup.