Commensurators of parabolic subgroups

of Coxeter groups

Author:
Luis Paris

Journal:
Proc. Amer. Math. Soc. **125** (1997), 731-738

MSC (1991):
Primary 20F55

DOI:
https://doi.org/10.1090/S0002-9939-97-03815-X

MathSciNet review:
1377001

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Abstract: Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by 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 in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

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Additional Information

**Luis Paris**

Affiliation:
Laboratoire de Topologie, Département de Mathématiques, Université de Bourgogne, U.M.R. 5584, B.P. 138, 21004 Dijon Cedex, France

Email:
lparis@satie.u-bourgogne.fr

DOI:
https://doi.org/10.1090/S0002-9939-97-03815-X

Received by editor(s):
October 17, 1995

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1997
American Mathematical Society