Reconstruction of Anosov flows from infinity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913555820838912 |
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| author | Baik, Hyungryul Wu, Chenxi Zhao, Bojun |
| author_facet | Baik, Hyungryul Wu, Chenxi Zhao, Bojun |
| contents | Every pseudo-Anosov flow $ϕ$ in a closed $3$-manifold $M$ gives rise to an action of $π_1(M)$ on a circle $S^{1}_{\infty}(ϕ)$ from infinity \cite{Fen12}, with a pair of invariant \emph{almost} laminations. From certain actions on $S^{1}$ with invariant almost laminations, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed $3$-manifolds. Our construction provides a geometry model for such flows and manifolds induced from $\mathcal{D} \times \mathcal{D}$, where $\mathcal{D}$ is the Poincaré disk with $\partial \mathcal{D}$ identified with $S^{1}_{\infty}(ϕ)$. In addition, our result applies to Cannon conjecture under the assumption that certain group-equivariant sphere-filling Peano curve exists, which offers a description of orientable quasigeodesic pseudo-Anosov flows in hyperbolic $3$-manifolds in terms of group actions on $\partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07634 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconstruction of Anosov flows from infinity Baik, Hyungryul Wu, Chenxi Zhao, Bojun Geometric Topology Every pseudo-Anosov flow $ϕ$ in a closed $3$-manifold $M$ gives rise to an action of $π_1(M)$ on a circle $S^{1}_{\infty}(ϕ)$ from infinity \cite{Fen12}, with a pair of invariant \emph{almost} laminations. From certain actions on $S^{1}$ with invariant almost laminations, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed $3$-manifolds. Our construction provides a geometry model for such flows and manifolds induced from $\mathcal{D} \times \mathcal{D}$, where $\mathcal{D}$ is the Poincaré disk with $\partial \mathcal{D}$ identified with $S^{1}_{\infty}(ϕ)$. In addition, our result applies to Cannon conjecture under the assumption that certain group-equivariant sphere-filling Peano curve exists, which offers a description of orientable quasigeodesic pseudo-Anosov flows in hyperbolic $3$-manifolds in terms of group actions on $\partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3}$. |
| title | Reconstruction of Anosov flows from infinity |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2407.07634 |