Narrow Results

A pair $(\alpha ,\beta )$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $\alpha$ is separating, then we call it as separating filling pair. In this paper, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\mathrm{\text{Mod}}(S_g)$ on the set of such filling pairs. Furthermore, we construct a Morse function ${{ℱ}}_g$ on the moduli space ${{ℳ}}_g$ which, for a given hyperbolic surface $X$, outputs the length of the shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function ${{ℱ}}_g$ is the same as the number of $\mathrm{\text{Mod}}(S_g)$-orbits of such filling pairs.

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).

There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.

It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

This paper investigates the geometric properties of random hyperbolic surfaces with respect to the Weil-Petersson measure. We describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volumes of the moduli spaces of hyperbolic surfaces of genus g as g → ∞. Then we apply these asymptotic estimates to study the geometric properties of random hyperbolic surfaces, such as the length of the shortest simple closed geodesic of a given combinatorial type.