Normalisateurs et groupes d'Artin-Tits de type sph\'erique

TL;DR

This paper investigates the structure of normalizers and quasi-centralizers of subgroups within spherical-type Artin-Tits groups, generalizing previous results and comparing these structures to Coxeter groups.

Contribution

It proves that in spherical-type Artin-Tits groups, the normalizer and commensurator of certain subgroups are equal and describes their structure using Coxeter group results.

Findings

Normalizers and commensurators are equal in spherical Artin-Tits groups.

The structure of quasi-centralizers is explicitly described.

Results generalize earlier work by Paris.

Abstract

Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the subgroup of $A_S$ generated by $X$. When $A_S$ is of spherical type, we prove that the normalizer and the commensurator of $A_X$ in $A_S$ are equal and are the product of $A_X$ by the quasi-centralizer of $A_X$ in $A_S$. Looking to the associated monoids $A_S^+$ and $A_X^+$, we describe the quasi-centralizer of $A_X^+$ in $A_S^+$ thanks to results in Coxeter groups. These two results generalize earlier results of Paris. Finaly, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin-Tits group and in the Coxeter group.