Counting geodesics on expander surfaces
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908830331305984 |
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| author | Dozier, Benjamin Sapir, Jenya |
| author_facet | Dozier, Benjamin Sapir, Jenya |
| contents | We study properties of typical closed geodesics on expander surfaces of high genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the Laplacian. Under an additional systole lower bound assumption, we show almost every geodesic of length much greater than $\sqrt{g}\log g$ is non-simple. And we prove almost every closed geodesic of length much greater than $g (\log g)^2$ is filling, i.e. each component of the complement of the geodesic is a topological disc. Our results apply to Weil-Petersson random surfaces, random covers of a fixed surface, and Brooks-Makover random surfaces, since these models are known to have uniform spectral gap asymptotically almost surely.
Our proof technique involves adapting Margulis' counting strategy to work at low length scales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07938 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting geodesics on expander surfaces Dozier, Benjamin Sapir, Jenya Geometric Topology Dynamical Systems Number Theory Probability We study properties of typical closed geodesics on expander surfaces of high genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the Laplacian. Under an additional systole lower bound assumption, we show almost every geodesic of length much greater than $\sqrt{g}\log g$ is non-simple. And we prove almost every closed geodesic of length much greater than $g (\log g)^2$ is filling, i.e. each component of the complement of the geodesic is a topological disc. Our results apply to Weil-Petersson random surfaces, random covers of a fixed surface, and Brooks-Makover random surfaces, since these models are known to have uniform spectral gap asymptotically almost surely. Our proof technique involves adapting Margulis' counting strategy to work at low length scales. |
| title | Counting geodesics on expander surfaces |
| topic | Geometric Topology Dynamical Systems Number Theory Probability |
| url | https://arxiv.org/abs/2304.07938 |