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

TL;DR

This paper investigates the dynamics of the SL(2,R) action on moduli space, establishing orbit closure and equidistribution results similar to unipotent flow theory, using measure classification and isolation properties.

Contribution

It introduces new methods to analyze orbit closures and equidistribution for SL(2,R) actions on moduli space, extending the understanding of their invariant manifolds.

Findings

Orbit closures are classified for the SL(2,R) action.

Equidistribution results are established for certain invariant measures.

Isolation properties of closed invariant manifolds are proved.

Abstract

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 [EMi2] and a certain isolation property of closed SL(2,R) invariant manifolds developed in this paper.