Isolation, equidistribution, and orbit closures for the SL(2,R) action on moduli space

We prove results about orbit closures and equidistribution for the SL(2, R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows.The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed SL(2, R) invariant manifolds developed in this paper.