Interval groups related to finite Coxeter groups, Part II

Barbara Baumeister, Derek Holt, George Neaime and Sarah Rees

Keywords

Coxeter groups, quasi-Coxeter elements, Artin groups, dual approach to Coxeter and Artin groups, generalised non-crossing partitions, Garside groups, Interval groups.

Status

Submitted for publication

Abstract

We consider interval groups related to proper 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. Type $D_n$ was proven in part I of this article. This 2nd article deals with the exceptional cases. Our techniques also reprove an analogous result for the Artin groups of the same type, 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. In addition, we prove using methods of Tits that the interval groups of proper quasi-Coxeter elements are not isomorphic to the Artin group of the same type, in the case of $D_n$ when $n$ is even or in any of the exceptional cases. In a subsequentarticle, we show that this result holds for type Dn for all n at least 4.

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