Coxeter covers of the classical Coxeter groups

TL;DR

This paper characterizes certain generalized Coxeter groups as semidirect products involving Abelian groups and classical Coxeter groups, extending previous results to a broader class of groups.

Contribution

It provides a new isomorphism classification for quotients of generalized Coxeter groups onto classical Coxeter groups, including cases with loops in the associated graphs.

Findings

C_Y(T) is isomorphic to A_{t,n} semidirect B_n or D_n

The structure depends on whether the graph T contains loops

The number of cycles in T determines t

Abstract

Let $C(T)$ be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either $B_n$ or $D_n$. Let $C_Y(T)$ be a natural quotient of $C(T)$, and if $C(T)$ is simply-laced (which means all the relations between the generators has order 2 or 3), $C_Y(T)$ is a generalized Coxeter group, too . Let $A_{t,n}$ be a group which contains $t$ Abelian groups generated by $n$ elements. The main result in this paper is that $C_Y(T)$ is isomorphic to $A_{t,n} \semidirect B_n$ or $A_{t,n} \semidirect D_n$, depends on whether the signed graph $T$ contains loops or not, or in other words C(T) is simply-laced or not, and $t$ is the number of the cycles in $T$. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.