Relative shapes of thick subsets of moduli space

Abstract

A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from $0$), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces {\it trivalent}. In this paper, in a first attempt to understand the ``shape'' of the subset ${\bf X}_g$ of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in $g$, with the set ${\bf Y}_g$ of trivalent surfaces. As our main result, we find that the set ${\bf X}_g\cap{\bf Y}_g$ is metrically ``sparse'' in ${\bf X}_g$ (where we equip $({\cal M}_g$ with either the Thurston or the Teichm\"uller metric).