2023/03/19 by Daniele Alessandrini, Alessandrini, Daniele, Sara Maloni +5 · 1 citation
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2303.10786
openalex publication_date 2023/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Anosov representations ρ of a hyperbolic group Γ into a semisimple Lie group G are known to admit cocompact domains of discontinuity in flag varieties G/Q, endowing the compact quotient manifolds Mρ with a (G,G/Q)-structure. In general the topology of Mρ can be quite complicated. In this article, we consider the case when Γ is the fundamental group of a closed (real or complex) hyperbolic manifold N and ρ is a deformation of a (twisted) lattice embedding Γ→ Isom(\mathbb H_\mathbb K) → G through Anosov representations. We prove that, in this situation, Mρ is alway a smooth fiber bundle over N. Determining the topology of the fiber seems hard in general. The second part of the paper focuses on the special case when N is a surface, ρ a quasi-Hitchin representation into Sp(4,\mathbb C), and Mρ is modelled on the space of complex Lagrangians in \mathbb C4. We show that, in this case, the fiber is homeomorphic to \mathbbCP2 \sharp \mathbbCP2.