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Convex cocompactness in pseudo-Riemannian hyperbolic spaces. (English) Zbl 1428.53078

Summary: Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups \(\mathrm{PO}(p,q)\) by considering their action on the associated pseudo-Riemannian hyperbolic space \(\mathbb {H}^{p,q-1}\) in place of the Riemannian symmetric space. Following work of T. Barbot and Q. Mérigot [Groups Geom. Dyn. 6, No. 3, 441–483 (2012; Zbl 1333.53106)] in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
20F55 Reflection and Coxeter groups (group-theoretic aspects)
22E40 Discrete subgroups of Lie groups
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
57S30 Discontinuous groups of transformations

Citations:

Zbl 1333.53106
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