Interval groups related to finite Coxeter groups I

Abstract

We elaborate presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. This is the only case of the infinite families of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call Carter generating set, and the relations are those defined by the related Carter diagram along with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. In a subsequent work, we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations. Alongside the proof of the main results of this paper, we establish important properties related to the dual approach to Coxeter and Artin groups