From arcs to curves: quadratic growth of 1-systems

Tarik Aougab; Jonah Gaster

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arXiv · 2508.05555

From arcs to curves: quadratic growth of 1-systems

Tarik Aougab·Jonah Gaster

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Abstract

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 $\chi$ grows quadratically in $|\chi|$. 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.

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Tarik Aougab, Jonah Gaster. 2025-08-07. From arcs to curves: quadratic growth of 1-systems. https://arxiv.org/abs/2508.05555

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