Small Filling Sets of Curves on a Surface

Consider a set of simple closed curves on a surface of genus g which fill the surface and which pairwise intersect at most once. We show that the asymptotic growth rate of the smallest number in such a set is 2 √ g as g → ∞. More generally, we give a precise asymptotic for filling sets of curves which pairwise intersect at most K ≥ 1 times. We then bound from below the cardinality of a filling set of systoles by g log(g) . The topological condition that a set of curves pairwise intersect at most once is thus quite far from the geometric condition that a set of curves can arise as systoles.