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

Let S be a surface of finite type; that is, a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we classify (globally) geodesic rays in the moduli space M(S) of Riemann surfaces, endowed with the Teichmüller metric, and we determine precisely how pairs of rays asymptote. We then use these results to relate two important but disparate topics in the study ofM(S): Teichmüller geometry and the DeligneMumford 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 constructing 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. Rays in M(S). A ray in a metric space X is a map r : [0,∞) → X which is locally an isometric embedding. In this paper we initiate the study of (globally) isometrically embedded rays inM(S). Among other things, we classify such rays, determine their asymptotics, classify almost geodesic rays, and work out the Tits angles between rays. We take as a model for our study the case of rays in locally symmetric spaces, as in the work of Borel, Ji, MacPherson and others; see [JM] for a summary. In [JM] it is explained how the continuous spectrum of any noncompact, complete Riemannian manifold M depends only on the geometry of its ends, and in some cases (e.g. when M is locally symmetric) the generalized eigenspaces can be parametrized by a compactification constructed from asymptote classes of certain rays. The spectral theory of M(S) ∗Both authors are supported in part by the NSF.