  We study the dynamics of the Teichmuller flow in the moduli space of Abelian
differentials (and more generally, its restriction to any connected component
of a stratum). We show that the (Masur-Veech) absolutely continuous invariant
probability measure is exponentially mixing for the class of Holder
observables. A geometric consequence is that the $\SL(2,\R)$ action in the
moduli space has a spectral gap.
