On irreducible, infinite, nonaffine Coxeter groups

Dongwen Qi¹

¹ Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210, U.S.A.

Fundamenta Mathematicae, Tome 193 (2007) no. 1, pp. 79-93

Résumé

The following results are proved: The center of any finite index subgroup of an irreducible, infinite, nonaffine Coxeter group is trivial; Any finite index subgroup of an irreducible, infinite, nonaffine Coxeter group cannot be expressed as a product of two nontrivial subgroups. These two theorems imply a unique decomposition theorem for a class of Coxeter groups. We also prove that the orbit of each element other than the identity under the conjugation action in an irreducible, infinite, nonaffine Coxeter group is an infinite set. This implies that an irreducible, infinite Coxeter group is affine if and only if it contains an abelian subgroup of finite index.

Détail

DOI : 10.4064/fm193-1-5

Keywords: following results proved center finite index subgroup irreducible infinite nonaffine coxeter group trivial finite index subgroup irreducible infinite nonaffine coxeter group cannot expressed product nontrivial subgroups these theorems imply unique decomposition theorem class coxeter groups prove orbit each element other identity under conjugation action irreducible infinite nonaffine coxeter group infinite set implies irreducible infinite coxeter group affine only contains abelian subgroup finite index