Small filling sets of curves on a surface

TL;DR

This paper investigates the minimal size of sets of simple closed curves on surfaces that fill the surface and have bounded pairwise intersections, revealing asymptotic growth rates and bounds related to genus and systoles.

Contribution

It establishes the asymptotic growth rate of minimal filling sets with bounded intersections and provides lower bounds for systolic filling sets, highlighting differences between topological and geometric conditions.

Findings

Asymptotic growth rate of minimal filling sets is 2√g/√K as g→∞.

Lower bound for systolic filling sets is g/ log(g).

Topological intersection conditions differ significantly from geometric systolic conditions.

Abstract

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus $g$ which fill and pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to \infty$ . We then bound from below the cardinality of a filling set of systoles by $g/\log(g)$. This illustrates that the topological condition that a set of curves pairwise intersect at most once is quite far from the geometric condition that such a set of curves can arise as systoles.