Filling systems of maximum size

Abstract

Let $S_g$ be a closed orientable surface of genus $g\ge 2$. A collection $\hat{I}\text{c◯}=\{{\hat{I}}_1^3,\dots ,{\hat{I}}_s^3\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that ${\hat{I}}_i^3$ and ${\hat{I}}_j^3$ are in minimal position, is called a \emph{filling system} or a \emph{filling} of $S_g$ if the complement $S_g\setminus \hat{I}\text{c◯}$ is a disjoint union of $b$ topological discs for some $b\ge 1$. The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on $S_g$ with $1\le b\le 2g-2$ boundary components is $2g+b-1$. Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with $1\le b\le g-2$ boundary components.

11 pages, 10 figures