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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2408.14041 |
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| _version_ | 1866912895335399424 |
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| author | Fitzhugh, Nick Schondorf, Aaron Shrestha, Sunrose Fallon, Sebastian Vander Ploeg Zeng, Thomas |
| author_facet | Fitzhugh, Nick Schondorf, Aaron Shrestha, Sunrose Fallon, Sebastian Vander Ploeg Zeng, Thomas |
| contents | A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. After a labelling of the squares by $\{1, \dots, n\}$, we can describe an STS with $n$ squares using two permutations $σ, τ\in S_n$, where $σ$ encodes how the squares are glued horizontally and $τ$ encodes how the squares are glued vertically. Hence, a previously considered natural model for STSs with $n$ squares is $S_n \times S_n$ with the uniform distribution. We modify this model to obtain a new one: We fix $α\in [0,1]$ and let $\mathcal{K}_{μ_n}$ be a conjugacy class of $S_n$ with at most $n^α$ cycles. Then $\mathcal{K}_{μ_n} \times S_n$ with the uniform distribution is a model for STSs with restricted horizontal gluings. Since horizontal cycles of the $σ$ permutation are related to the number of maximal horizontal cylinders, this new model serves as a random model for STSs with at most $n^α$ maximal horizontal cylinders. We deduce the asymptotic (as $n$ grows) number of components, genus distribution, most likely stratum and set of holonomy vectors of saddle connections for random STSs in this new model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A model for horizontally restricted random square-tiled surfaces Fitzhugh, Nick Schondorf, Aaron Shrestha, Sunrose Fallon, Sebastian Vander Ploeg Zeng, Thomas Geometric Topology Probability A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. After a labelling of the squares by $\{1, \dots, n\}$, we can describe an STS with $n$ squares using two permutations $σ, τ\in S_n$, where $σ$ encodes how the squares are glued horizontally and $τ$ encodes how the squares are glued vertically. Hence, a previously considered natural model for STSs with $n$ squares is $S_n \times S_n$ with the uniform distribution. We modify this model to obtain a new one: We fix $α\in [0,1]$ and let $\mathcal{K}_{μ_n}$ be a conjugacy class of $S_n$ with at most $n^α$ cycles. Then $\mathcal{K}_{μ_n} \times S_n$ with the uniform distribution is a model for STSs with restricted horizontal gluings. Since horizontal cycles of the $σ$ permutation are related to the number of maximal horizontal cylinders, this new model serves as a random model for STSs with at most $n^α$ maximal horizontal cylinders. We deduce the asymptotic (as $n$ grows) number of components, genus distribution, most likely stratum and set of holonomy vectors of saddle connections for random STSs in this new model. |
| title | A model for horizontally restricted random square-tiled surfaces |
| topic | Geometric Topology Probability |
| url | https://arxiv.org/abs/2408.14041 |