Small filling sets of curves on a surface

Description/Abstract

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\sqrt{g} as g goes to infinity. More generally, we give a precise asymptotic for filling sets of curves which pairwise intersect at most K times, where K is greater than equal to 1. 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.