From arcs to curves: quadratic growth of 1-systems

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.