Ergodic Properties of the Horocycle Flow and Classification of Fuchsian Groups

The paper is devoted to a study of the basic ergodic properties (ergodicity and conservativity) of the horocycle ow on surfaces of constant negative curvature with respect to the Liouville invariant measure. We give several criteria for ergodicity and conservativity and connect them with the classiication of the associated Fuchsian groups. Special attention is given to covering surfaces. In particular, we show that normal subgroups of divergent type Fuchsian groups provide natural examples for the strictness of a number of inclusions in the classiication of Fuchsian groups. The paper in a sense complements and continues earlier work by A. N. Starkov \Fuchsian groups from the dynamical viewpoint" published in J. The horocycle and the geodesic ows on surfaces of constant negative curvature are the basic examples of ows on homogeneous spaces. The general theory of such ows so far mostly deals with the coonite volume case (e.g., see St99]). In this situation establishing ergodicity (conservativity being self-evident) is just the rst step before studying much more subtle properties. In particular, ergodicity of both the horocycle and the geodesic ows with respect to the Liouville measure on nite volume surfaces has been known for a long time. In the innnite volume case even such basic ergodic properties as ergodicity and conser-vativity of the Liouville measure become non-trivial. For the geodesic ow this problem is completely resolved by the Hopf{Tsuji{Sullivan theorem Su81], according to which ergodicity is equivalent to conservativity of the geodesic ow, and both are equivalent to recurrence of the Brownian motion on the surface (the associated Fuchsian group being of divergent type). However, until recently little was known about the ergodic properties of the horocycle ow in the innnite volume case. The paper St95] by Starkov resuscitated this problem. In particular, he conjectured that ergodicity of the horocycle ow is equivalent to ergodicity of the boundary action of the associated Fuchsian group. Inspired by this conjecture, Babillot and Ledrappier BL98] proved that the horocycle ow is ergodic on all Z d-covers of compact surfaces (independently of the degree of the cover, in contrast to the geodesic ow case); another proof of this fact was later given by Pollicott Po98]. Here we prove Starkov's conjecture in full generality. In fact, we give a complete description of the ergodic components of the horocycle ow in terms of