There is an action of SL(2,R) on moduli space of a compact Riemann surface which is expected to have similar rigidity properties as in the theory of unipotent flows on homogeneous spaces. In this talk we will review basic definitions and properties of these actions, describing some motivating problems. We will also give and overview on the progress made on these problems over the past 40 years or so.
Specifically, we will discuss a joint work with Eskin and Mirzakhani in which orbits closures of this action are studied. The main ingredients are a measure classification theorem of Eskin and Mirzakhani, and a certain avoidance principle which we develop using some ideas of Margulis.