Relative shapes of thick subsets of moduli space

TL;DR

This paper investigates the geometric structure of a subset of moduli space consisting of surfaces with filling systoles, comparing it to trivalent surfaces and finding it to be metrically sparse.

Contribution

It provides the first comparison of the shape of the subset of surfaces with filling systoles to trivalent surfaces in moduli space, revealing their sparse metric relationship.

Findings

The set of surfaces with filling systoles is metrically sparse within the subset of trivalent surfaces.

The comparison is made asymptotically in genus g, using Thurston and Teichmüller metrics.

The results offer new insights into the geometric complexity of moduli space subsets.

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).