Fiber bundles associated with Anosov representations

Anosov representations of hyperbolic groups form a rich class of representations that are closely related to geometric structures on closed manifolds. Any Anosov representation admits cocompact domains of discontinuity in flag varieties [GW12, KLP18] endowing the compact quotient manifolds with a -structure. In general, the topology of can be quite complicated. In this article, we will focus on the special case when is a the fundamental group of a closed (real or complex) hyperbolic manifold N and is a deformation of a (twisted) lattice embedding through Anosov representations. In this case, we prove that is a smooth fiber bundle over N, and we describe the structure group of this bundle and compute its invariants. This theorem applies in particular to most representations in higher rank TeichmÜller spaces, as well as convex divisible representations, AdS-quasi-Fuchsian representations and -convex cocompact representations. Even when is a fiber bundle, it is often very difficult to determine the fiber. In the second part of the paper, we focus on the special case when N is a surface, a quasi-Hitchin representation into, and carries a -structure. We show that in this case the fiber is homeomorphic to. This fiber bundle is of particular interest in the context of possible generalizations of Bers' double uniformization theorem in the context of higher rank TeichmÜller spaces, since for Hitchin-representations it contains two copies of the locally symmetric space associated to. Our result uses the classification of smooth -manifolds, the study of the -orbits of and the identification of with the space of (unlabelled) regular ideal hyperbolic tetrahedra and their degenerations.