Commensurators of parabolic subgroups of Coxeter groups

Abstract

Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup of $W$ generated by $X$ is denoted by $W_X$ and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of $W_X$ in $W$ is the subgroup of $w$ in $W$ such that $w{W}_X{w}^{-1}\cap W_X$ has finite index in both $W_X$ and $w{W}_X{w}^{-1}$. The subgroup $W_X$ can be decomposed in the form $W_X=W_{X^0}\cdot W_{X^{\infty }}\simeq W_{X^0}\times W_{X^{\infty }}$ where $W_{X^0}$ is finite and all the irreducible components of $W_{X^{\infty }}$" > are infinite. Let $Y^{\infty }$ be the set of $t$ in $S$ such that $m_{s,t}=2$" > for all $s\in X^{\infty }$. We prove that the commensurator of $W_X$ is $W_{Y^{\infty }}\cdot W_{X^{\infty }}\simeq W_{Y^{\infty }}\times W_{X^{\infty }}$. In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and $W_X$ is its own commensurator if and only if $X^0=Y^{\infty }$.

Plain tex version, 9 pages no figures

Cited by in corpus (2)

• On centralizers of parabolic subgroups in Coxeter groups
• Quasi-Isometries for certain Right-Angled Coxeter Groups