Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915093080440832 |
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| author | Fenley, Sergio R. Potrie, Rafael |
| author_facet | Fenley, Sergio R. Potrie, Rafael |
| contents | Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation $\mathcal{G}_L$ induced by $\mathcal{G}$ in the universal cover $L$ of any leaf of $\mathcal{F}_1$ or $\mathcal{F}_2$ has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_05176 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic Fenley, Sergio R. Potrie, Rafael Geometric Topology Differential Geometry Dynamical Systems Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation $\mathcal{G}_L$ induced by $\mathcal{G}$ in the universal cover $L$ of any leaf of $\mathcal{F}_1$ or $\mathcal{F}_2$ has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves. |
| title | Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic |
| topic | Geometric Topology Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2310.05176 |