From arcs to curves: quadratic growth of 1-systems
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915433865543680 |
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| author | Aougab, Tarik Gaster, Jonah |
| author_facet | Aougab, Tarik Gaster, Jonah |
| contents | We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $χ$ grows quadratically in $|χ|$. This resolves a longstanding question of Farb-Leininger, up to multiplicative constants. Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of \textit{almost nibs}, \textit{flowers}, and \textit{stem systems} in order to account for how certain polygons built from pairs of curves in the collection distribute their area over the surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From arcs to curves: quadratic growth of 1-systems Aougab, Tarik Gaster, Jonah Geometric Topology Combinatorics 57M15 05B99 57M50 We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $χ$ grows quadratically in $|χ|$. This resolves a longstanding question of Farb-Leininger, up to multiplicative constants. Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of \textit{almost nibs}, \textit{flowers}, and \textit{stem systems} in order to account for how certain polygons built from pairs of curves in the collection distribute their area over the surface. |
| title | From arcs to curves: quadratic growth of 1-systems |
| topic | Geometric Topology Combinatorics 57M15 05B99 57M50 |
| url | https://arxiv.org/abs/2508.05555 |