Year: 2024
Type: article
Abstract: We investigate representations of Coxeter groups into \mathrm{GL}(n,\mathbb{R}) as geometric reflection groups which are convex cocompact in the projective space \mathbb{P}(\mathbb{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 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 Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in \mathbb{P}(\mathbb{R}^{n}) developed by Danciger–Guéritaud–Kassel (2024).
DOI: 10.4171/jems/1539
Institutions: The University of Texas at Austin, Centre National de la Recherche Scientifique, Institut de Recherche Mathématique Avancée, Université de Strasbourg, Université Paris-Saclay, Institut des Hautes Études Scientifiques, Seoul National University, Sungkyunkwan University, Institut de recherche mathématique de Rennes, Université de Rennes
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funders: National Science Foundation, Alfred P. Sloan Foundation, Centre Henri Lebesgue, European Commission, National Research Foundation, Deutsche Forschungsgemeinschaft, Agence Nationale de la Recherche, National Research Foundation of Korea, Ministry of Science and ICT, South Korea, Engineering and Physical Sciences Research Council, H2020 European Research Council, FP7 Ideas: European Research Council
Awards: Dynamics and geometric structures, G1669854408, RNMS: Geometric structures and representation varieties, Deformation Spaces of Geometric Structures, Discrete Groups and Geometric Structures, G3257718068, Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation, Isaac Newton Institute for Mathematical Sciences, G3815200236, G3880489783, G434337789, G4421442999, Centre Européen pour les Mathématiques, la Physique et leurs Interactions, CAREER: Locally Homogeneous Geometric Manifolds and Their Moduli Spaces, G4947591155, G5128091834, Spaces of geometric structures via geometric transitions, G5688119252, Deformation Spaces of Geometric Structures, Mathematical Sciences Research Institute (MSRI), RNMS: Geometric Structures and Representation Varieties, G8981253443
OpenAlex ID: W3127829407