2025/09/30 by Suhyoung Choi, Choi, Suhyoung
Mathematics · Physics and Astronomy · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2509.26117
openalex publication_date 2025/09/30 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
Let N be a manifold of dimension m with a flat vector bundle given by a representation ρ:π1(N) → GL(n, R) where π1(N) is finitely generated. The holonomy group ρ is a k-partially hyperbolic holonomy representation if the flat bundle pulled back over the unit tangent bundle of a sufficiently large compact submanifold of N splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are k-dimensional with k < n/2. Suppose that each element of ρ(π1(N)) has an eigenvalue of norm 1, or, alternatively, ρ has some singular values of subexponential growth in terms of word length. We show that ρ is a P-Anosov representation for a parabolic subgroup P of GL(n, R) if and only if ρ is a partially hyperbolic representation. We are going to primarily employ representation theory techniques. As an application, we will show that the equivalence holds when N is a complete affine n-manifold, and ρ is a linear part of the holonomy representation. This had never been done over the full general linear group.