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 {\em trivalent}. In this paper, in a first attempt to understand the "shape" of the subset $\ts$ of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set $\tri$ of trivalent surfaces. As our main result, we find that the set $\ts \cap \tri$ is metrically "sparse" in $\ts$ (where we equip $\moduli$ with either the Thurston or the Teichm\"uller metric).