The horocycle flow on the unit tangent bundle of a surface of constant negative curvature is the unit speed translation along the stable leaves of the geodesic flow. For compact or finite volume surfaces, its qualitative (as well as, to a good extent, its quantitative) ergodic properties are well-understood. The situation is more delicate and less understood when the surface has infinite volume. In this talk, I will focus on the case of Abelian covers of compact hyperbolic surfaces, in particular I will discuss a joint work with Livio Flaminio which describes the mixing asymptotics of horocycle flow. An analogous result for the geodesic flow was proved recently by Dolgopyat, Nandori and Pène by a different method based on symbolic dynamics and the spectral theory of transfer operators. Our approach relies instead on the representation theory of SL_2(R); the crucial difference with the finite volume case is the absence of a spectral gap. As a byproduct, we recover a result by Jakobson, Naud and Soares on the equidistribution of the eigenvalues of the Casimir operator near zero for increasing sequences of finite Abelian covers.

https://indico.math.cnrs.fr/event/9181/