Interval groups related to finite Coxeter groups, Part II

TL;DR

This paper characterizes the structure of interval groups related to quasi-Coxeter elements in finite Coxeter groups, revealing new Garside structures and classifying isomorphisms with Artin groups across different types.

Contribution

It provides a complete presentation of these interval groups, introduces a new Garside structure in non-simply laced cases, and classifies when they are isomorphic to Artin groups.

Findings

Interval groups are quotients of Artin groups by twisted cycle commutators.

A new Garside structure is discovered for non-simply laced cases.

Interval groups of proper quasi-Coxeter elements are not isomorphic to Artin groups in certain types.

Abstract

We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4-cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non-simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi-Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of $D_n$ when $n$ is even or in any of the exceptional cases. In [BHNR22], we show using different methods that this result holds for type $D_n$ for all $n \geq 4$.