## Abstract

Let *S* = *S _{g, p}
* be a compact, orientable surface of genus

*g*with

*p*punctures and such that

*d*(

*S*) ≔ 3

*g*– 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.

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