Convex cocompactness for Coxeter groups

Jeffrey Danciger (1) , François Guéritaud (2, 3) , Fanny Kassel (3, 4, 5) , Gye-Seon Lee (6) , Ludovic Marquis (7)

1 University of Texas at Austin [Austin]
2 IRMA - Institut de Recherche Mathématique Avancée
3 CNRS - Centre National de la Recherche Scientifique
4 IHES - Institut des Hautes Études Scientifiques
5 Université Paris-Saclay
6 Department of Mathematics, Sungkyunkwan University
7 IRMAR - Institut de Recherche Mathématique de Rennes

We investigate representations of Coxeter groups into GL(n,R) as geometric reflection groups which are convex cocompact in the projective space P(R^n). We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic, irreducible Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic irreducible Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in P(R^n) developed in arXiv:1704.08711.

Domaines: Mathématiques [math]

Jeffrey Danciger, François Guéritaud, Fanny Kassel, Gye-Seon Lee, Ludovic Marquis. Convex cocompactness for Coxeter groups. 2022. ⟨hal-03172107⟩