Isomorphism and non-isomorphism for interval groups of type D_n

TL;DR

This paper investigates the structure of interval groups related to Coxeter groups of type D_n, providing new presentations and demonstrating non-isomorphism with Artin groups, illustrating broader group-theoretic techniques.

Contribution

It introduces alternative combinatorial presentations for interval groups of type D_n and proves they are not isomorphic to corresponding Artin groups.

Findings

Interval groups of proper quasi-Coxeter elements are not isomorphic to Artin groups of the same type.

New combinatorial presentations for these interval groups are derived.

Techniques used can be applied to study properties of infinite families of groups.

Abstract

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of $W(D_n)$ cannot be isomorphic to the Artin group of type $D_n$. While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational observation can be used to derive properties of all groups within an infinite family.