The even isomorphism theorem for Coxeter groups

Abstract:

Coxeter groups have presentations $\langle S :(st)^{m_{st}}\forall s,t\in S \rangle$ where for all $s,t\in S$, $m_{st}\in \{1,2,\ldots ,\infty \}$, $m_{st}=m_{ts}$ and $m_{st}=1$ if and only if $s=t$. A fundamental question in the theory of Coxeter groups is: Given two such “Coxeter" presentations, do they present the same group? There are two known ways to change a Coxeter presentation, generally referred to as twisting and simplex exchange. We solve the isomorphism question for Coxeter groups with an even Coxeter presentation (one in which $m_{st}$ is even or $\infty$ when $s\ne t$). More specifically, we give an algorithm that describes a sequence of twists and triangle-edge exchanges that either converts an arbitrary finitely generated Coxeter presentation into a unique even presentation or identifies the group as a non-even Coxeter group. Our technique can be used to produce all Coxeter presentations for a given even Coxeter group.