Sharp bounds for 1-systems (Tarik Aougab)

Abstract:  On a closed orientable surface of genus $g$, a 1-system is a collection of pairwise non-homotopic simple closed curves that pairwise intersect at most once. Obtaining bounds on the maximum size of a 1-system has proved to be a surprisingly hard problem. Constructions with roughly $g^2$ curves have been known for the last few decades, but upper bounds are trickier: in 2012, Malestein-Rivin-Theran produced an upper bound that is exponential in $g$. Przytycki in 2014 improved this to a bound that is $O(g^3)$, and in 2018 Josh Greene achieved an upper bound that behaves like $g^2 \log(g)$.

Our main result is a quadratic-in-$g$ upper bound, resolving the problem up to explicit multiplicative constants. We achieve this by choosing an appropriate hyperbolic metric and paying careful attention to how certain polygons formed by curves in the 1-system distribute their area over the surface. This represents joint work with Jonah Gaster.