Filling sets of curves on punctured surfaces

TL;DR

This paper investigates the minimal size of sets of simple closed curves on punctured surfaces that fill the surface and have bounded pairwise intersections, analyzing growth rates and hyperbolic systolic representations.

Contribution

It establishes growth orders for the size of filling curve sets with bounded intersections, highlighting differences between even and odd intersection bounds and considering hyperbolic systoles.

Findings

Derived growth orders for even k

Identified different behaviors for odd k

Connected filling sets to systolic curves on hyperbolic surfaces

Abstract

We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for odd k the orders of growth behave differently. We also study the corresponding questions when one requires that the curves be represented as systoles on hyperbolic complete finite area surfaces.