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

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

2008-03-02

arxiv.org/abs/0803.0141v2

Mathematics

Geometric Topology

Final revision. 42 pages. To appear in J. Diff. Geom

Scientific paper

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.

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