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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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Convex co‐compact representations of 3‐manifold groups
Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.
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- PAR ID:
- 10508833
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 17
- Issue:
- 2
- ISSN:
- 1753-8416
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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