Word problem and parabolic subgroups in Dyer groups

Abstract

AbstractOne 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.