Teichmuller geometry of moduli space, I: Distance minimizing rays and the Deligne-Mumford compactification

TL;DR

This paper reconstructs the Deligne-Mumford compactification of moduli space from the intrinsic Teichmüller metric by classifying geodesic rays and analyzing their asymptotic behavior.

Contribution

It introduces a method to derive the Deligne-Mumford compactification solely from the metric geometry of Teichmüller space, including classifying geodesic rays and constructing an iterated EDM ray space.

Findings

Classified geodesic rays in Teichmüller space.

Established a correspondence between the iterated EDM ray space and the Deligne-Mumford compactification.

Provided a metric-based reconstruction of the compactification.

Abstract

Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\M(S)$ of Riemann surfaces: Teichm\"{u}ller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of $\M(S)$ endowed with the Teichm\"{u}ller metric. We do this by first classifying (globally) geodesic rays in $\M(S)$ and determining precisely how pairs of rays asymptote. We construct an "iterated EDM ray space" functor, which is defined on a quite general class of metric spaces. We then prove that this functor applied to $\M(S)$ produces the Deligne-Mumford compactification.