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

Benson Farb; Howard Masur

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arXiv · 0803.0141

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

Benson Farb·Howard Masur

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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: Teichmüller 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 Teichmüller 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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BibTeX RIS

Benson Farb, Howard Masur. 2010-06-18. Teichmuller geometry of moduli space, I: Distance minimizing rays and the Deligne-Mumford compactification. https://arxiv.org/abs/0803.0141

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