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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.26117 |
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| _version_ | 1866912905459400704 |
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| author | Choi, Suhyoung |
| author_facet | Choi, Suhyoung |
| contents | 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, $ρ$ 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 $\mathrm{GL}(n, \mathbf{R})$ 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partially hyperbolic flows on flat vector bundles with an application to complete affine manifolds Choi, Suhyoung Geometric Topology Representation Theory 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, $ρ$ 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 $\mathrm{GL}(n, \mathbf{R})$ 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 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. |
| title | Partially hyperbolic flows on flat vector bundles with an application to complete affine manifolds |
| topic | Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2509.26117 |