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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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Coarse Distance from Dynamically Convex to Convex
Abstract Chaidez and Edtmair have recently found the first examples of dynamically convex domains in $$\mathbb{R}^{4}$$ that are not symplectomorphic to convex domains, answering a long-standing open question. In this paper, we discover new examples of such domains without referring to Chaidez–Edtmair’s methods. We show a stronger result: that these domains are arbitrarily far from the set of convex domains in $$\mathbb{R}^{4}$$ with respect to the coarse symplectic Banach–Mazur distance.
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- Award ID(s):
- 1926686
- PAR ID:
- 10610540
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 12
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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