Teichmüller geometry of moduli space, II: M(S) seen from far away

where K ≥ 1 is the least number such that there is a K-quasiconformal mapping between the marked structures X1 and X2. The mapping class group Mod(S) acts properly discontinuously and isometrically on Teich(S), thus inducing a metric dM(S)(·, ·) on the quotient moduli space M(S) := Teich(S)/Mod(S). Let π : Teich(S) → M(S) be the natural projection. The goal of this paper is to build an “almost isometric” simplicial model for M(S), from which we will determine the tangent cone at infinity of M(S). In analogy with the case of locally symmetric spaces, this can be viewed as a step in building a “reduction theory” for the action of Mod(S) on Teich(S). Other results in this direction can be found in [Le].