Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Abstract

Abstract. 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 PO(p, q) by considering their action on the associated pseudoRiemannian hyperbolic space H p,q−1 in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

Affine actions with Hitchin linear part

“…For recent examples of Anosov representations of right-angled Coxeter groups, see [DGK17,DGK18a]. Hitchin representations are the main examples of Anosov representations of concern in this paper.…”

Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry

“…it is contained and convex in some affine chart). We follow the work of [DGK18] and say that a discrete group Γ ⊂ PO(p, q + 1) is H p,qconvex cocompact if it acts properly discontinuously and cocompactly on a closed properly convex set Ω ⊂ H p,q with non empty interior such that the intersection of the closure Ω ⊂ RP p+q with ∂H p,q does not contain any non trivial line segment.…”

“…• When m is even (resp. odd), Hitchin representations in PO(m + 1, m) are H m,m -convex cocompact (resp.H m+1,m−1 -convex cocompact) [DGK18].…”

“…But the rigidity theorem is trivial in this case. It is shown in [DGK18] that it is always possible to choose Ω strictly convex with C 1 boundary.…”

“…The goal of this paper is to understand the geometry of these limit sets for a specific class of subgroups of PO(p, q + 1) introduced by J. Danciger, F. Guéritaud and F. Kassel in [DGK18] that are related to pseudo-Riemannian hyperbolic geometry. More precisely, we will define a geometric quantity associated to the limit set which we call the pseudo-Riemannian Hausdorff dimension (its generalizes the Hausdorff dimension), and show that it is equal to a notion of critical exponent in pseudo-Riemannian hyperbolic geometry.…”