Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914350498840576 |
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| author | Jeffreys, Luke Matheus, Carlos |
| author_facet | Jeffreys, Luke Matheus, Carlos |
| contents | Using algorithms implicit in the classification of $SL(2,\mathbb{Z})$-orbits of primitive origamis in the stratum $\mathcal{H}(2)$ due to Hubert-Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $\mathcal{H}(2)$ is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in $\mathcal{H}(4)$ and $\mathcal{H}(6),$ we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form $O(N^{2/3}\log N)$, where $N$ is the size of the orbit graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$ Jeffreys, Luke Matheus, Carlos Geometric Topology 32G15, 30F30, 30F60 (Primary) 05C12 (Secondary) Using algorithms implicit in the classification of $SL(2,\mathbb{Z})$-orbits of primitive origamis in the stratum $\mathcal{H}(2)$ due to Hubert-Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $\mathcal{H}(2)$ is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in $\mathcal{H}(4)$ and $\mathcal{H}(6),$ we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form $O(N^{2/3}\log N)$, where $N$ is the size of the orbit graph. |
| title | Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$ |
| topic | Geometric Topology 32G15, 30F30, 30F60 (Primary) 05C12 (Secondary) |
| url | https://arxiv.org/abs/2502.10358 |