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Let S = Sg, p be a compact, orientable surface of genus g with p punctures and such that d(S) ≔ 3g – 3 + p > 0. The mapping class group Mod S acts properly discontinuously on the Teichmüller space of marked hyperbolic structures on S. The resulting quotient ℳ(S) is the moduli space of isometry classes of hyperbolic surfaces. We provide a version of precise reduction theory for finite index subgroups of Mod S , i.e., a description of exact fundamental domains (for the thin part of ). As an application we show that the asymptotic cone of the moduli space ℳ(S) endowed with any metric in the bi-Lipschitz class of the Teichmüller metric is bi-Lipschitz equivalent to the Euclidean cone over the finite simplicial (orbi-) complex , where of S is the complex of curves of S. We also show that if d(S) ≧ 2, then ℳ(S) does not admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the bi-Lipschitz class of the Teichmüller metric. These two applications confirm conjectures of B. Farb.

Reduction theory for mapping class groups and applications to moduli spaces