Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,\mathbb{R})$ action on moduli space


We prove results about orbit closures and equidistribution for the $\mathrm{SL}(2,\mathbb{R})$ 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 $\mathrm{SL}(2,\mathbb{R})$ invariant manifolds developed in this paper.


Alex Eskin

University of Chicago, Chicago, IL

Maryam Mirzakhani

Stanford University, Stanford, CA

Amir Mohammadi

University of Texas, Austin, TX