Convex cocompactness for Coxeter groups

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).

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Publication Details

Journal
Journal of the European Mathematical Society
Published
2024-11-28
DOI
https://doi.org/10.4171/jems/1539
Citations
10
Primary Topic
Geometric and Algebraic Topology
Type
article
Field-Weighted Citation Impact
8.10

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article

Convex cocompactness for Coxeter groups

Gye‐Seon Lee, François Guéritaud, Ludovic Marquis, Fanny Kassel et al.
10 citations
Journal of the European Mathematical Society
Geometric and Algebraic Topology
8.10
article

Convex cocompactness for Coxeter groups

Gye‐Seon Lee, François Guéritaud, Ludovic Marquis, Fanny Kassel, Jeffrey Danciger
article en
10 citations

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).

Journal of the European Mathematical SocietyVol. 27(1)
Centre National de la Recherche Scientifique (FR), Seoul National University (KR), Université Paris-Saclay (FR), Institut de Recherche Mathématique Avancée (FR), Institut de recherche mathématique de Rennes (FR), Université de Rennes (FR), Université de Strasbourg (FR), Sungkyunkwan University (KR), Institut des Hautes Études Scientifiques (FR), The University of Texas at Austin (US)
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
Openalex Percentile: Top 2%
Geometric and Algebraic Topology
8.10
DOIOpen Access PDFOpenAlex
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