Word problem and parabolic subgroups in Dyer groups

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.

The spherical growth series of Dyer groups

“…The statements regarding syllabic length were proved in [13, Proposition 2.8]. Hence we know that there exists a unique such that for all we have: …”

“…Let be a reduced syllabic representative for g . We know by [13, Lemma 2.5] that , hence, by Corollary 3.3, …”

“…Let be a reduced syllabic representative for h . We know from [13, Lemma 2.5] that . But, if , then g is not X -minimal, and if , then g is not Y -minimal.…”