Partially hyperbolic flows on flat vector bundles with an application to complete affine manifolds
Suhyoung Choi
TL;DR
This work establishes a precise equivalence between partially hyperbolic representations in flat vector bundles and $P$-Anosov representations for the full general linear group ${\mathsf{GL}}(n,\mathbb{R})$, under eigenvalue-modulus constraints or uniform singular-value bounds, with the index $k$ satisfying $k < n/2$. By developing a bundle-oriented notion of partial hyperbolicity and connecting it to the dynamical contraction/expansion along the geodesic flow, the authors show that a holonomy representation $\rho: \pi_1(N) \to {\mathsf{GL}}(n,\mathbb{R})$ is $P_{\theta}$-Anosov for a parabolic $P_{\theta}$ (where $\theta$ contains $\log\lambda_k - \log\lambda_{k+1}$) if and only if it is partially hyperbolic in a bundle sense. The analysis relies on coarse geometric tools, Gromov flow spaces, and dominance/dynamics in the Benoist cone, and it extends prior Anosov-type results beyond orthogonal groups. As an application, the main result holds for complete affine $n$-manifolds where the linear part of the holonomy acts via $\rho$, connecting to Goldman–Labourie’s hyperbolic-flow perspective and recent higher-dimensional affine-holonomy work. The findings advance understanding of the interplay between hyperbolic dynamics and linear representations, with potential implications for the structure of affine manifolds and broader geometric group theory. The approach provides a framework to transfer dynamical properties across the holonomy representation to yield a robust partial hyperbolicity structure in the flat bundle setting, and it clarifies when Anosov properties can be realized in full ${\mathsf{GL}}(n,\mathbb{R})$-settings.
Abstract
Let $N$ be a manifold of dimension $m$ with a flat vector bundle given by a representation $ρ:π_1(N) \rightarrow \mathrm{GL}(n, \mathbf{R})$ where $π_1(N)$ 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 $ρ(π_1(N))$ has an eigenvalue of norm $1$, or, alternatively, each element of it has a singular value that is uniformly bounded above and below. We show that $ρ$ is a $P$-Anosov representation for a parabolic subgroup $P$ of $\mathrm{GL}(n, \mathbf{R})$ 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 this 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.