Imaginary cones and limit roots of infinite Coxeter groups

Dyer, Matthew; Hohlweg, Christophe et Ripoll, Vivien (2016). « Imaginary cones and limit roots of infinite Coxeter groups ». Mathematische Zeitschrift.

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Résumé

Let (W; S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular,we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.

Type:	Article de revue scientifique
Mots-clés ou Sujets:	Coxeter group, root system, roots, limit point, accumulation set, elementary roots, small roots.
Unité d'appartenance:	Faculté des sciences > Département de mathématiques
Déposé par:	Christophe Hohlweg
Date de dépôt:	12 mai 2016 12:17
Dernière modification:	30 mai 2016 17:17
Adresse URL :	http://archipel.uqam.ca/id/eprint/8444

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