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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).more » « lessFree, publicly-accessible full text available February 5, 2026
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Properly discontinuous actions of a surface group by affine automorphisms of ℝ^d were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in 𝖲𝖮(n,n−1), so that the affine action is by isometries of a flat pseudo-Riemannian metric on ℝ^d of signature (n,n−1). Here, the translational part determines a deformation of the linear part into 𝖯𝖲𝖮(n,n)-Hitchin representations and the crucial step is to show that such representations are not Anosov in 𝖯𝖲𝖫(2n,ℝ) with respect to the stabilizer of an n-plane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudo-Riemannian hyperbolic space of signature (n,n−1) by a 𝖯𝖲𝖮(n,n)-Hitchin representation fails to be properly discontinuous.more » « less
