Functorial Free Group from Anosov Representations on Bundles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914175486263296 |
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| author | Gongopadhyay, Krishnendu Nayak, Tathagata |
| author_facet | Gongopadhyay, Krishnendu Nayak, Tathagata |
| contents | Let $ρ: Γ\to G$ be an Anosov representation, with $Γ$ a word hyperbolic group and $G$ a semisimple Lie group. Previous works (Guichard--Wienhard, Kapovich--Leeb--Porti, and Carvajales--Stecker) constructed an open domain of discontinuity $Ω_ρ\subset G/H$, where $H$ is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous $Γ$-action via $ρ$ to the space of connections on the pullbacks of the tangent bundle over $Ω_ρ$.
When $Ω_ρ$ is a complex curve, we show that the $Γ$-action is properly discontinuous on the union of Higgs bundle structures associated with the $(1,0)$ part of the complexified pullback bundles. We further construct a free abelian group $F^{ab}$ generated by these holomorphic line bundles and induce a topoogical structure on it, so that $ρ(Γ)$ acts properly discontinuously on $F^{ab} \setminus \{\mathrm{id}\}$. This free abelian group is well-defined up to isomorphism over the character variety of Zariski dense Anosov representations. Finally, we endow the space of Anosov representations with a categorical structure compatible with $Ω_ρ$ and construct a natural functor to the category of abelian groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_17251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functorial Free Group from Anosov Representations on Bundles Gongopadhyay, Krishnendu Nayak, Tathagata Geometric Topology Differential Geometry Group Theory Representation Theory Primary 22F30, Secondary 22E46, 22E15, 22E40, 22F50, 57R22, 18F15, 55R99, 32L05 Let $ρ: Γ\to G$ be an Anosov representation, with $Γ$ a word hyperbolic group and $G$ a semisimple Lie group. Previous works (Guichard--Wienhard, Kapovich--Leeb--Porti, and Carvajales--Stecker) constructed an open domain of discontinuity $Ω_ρ\subset G/H$, where $H$ is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous $Γ$-action via $ρ$ to the space of connections on the pullbacks of the tangent bundle over $Ω_ρ$. When $Ω_ρ$ is a complex curve, we show that the $Γ$-action is properly discontinuous on the union of Higgs bundle structures associated with the $(1,0)$ part of the complexified pullback bundles. We further construct a free abelian group $F^{ab}$ generated by these holomorphic line bundles and induce a topoogical structure on it, so that $ρ(Γ)$ acts properly discontinuously on $F^{ab} \setminus \{\mathrm{id}\}$. This free abelian group is well-defined up to isomorphism over the character variety of Zariski dense Anosov representations. Finally, we endow the space of Anosov representations with a categorical structure compatible with $Ω_ρ$ and construct a natural functor to the category of abelian groups. |
| title | Functorial Free Group from Anosov Representations on Bundles |
| topic | Geometric Topology Differential Geometry Group Theory Representation Theory Primary 22F30, Secondary 22E46, 22E15, 22E40, 22F50, 57R22, 18F15, 55R99, 32L05 |
| url | https://arxiv.org/abs/2507.17251 |