Abstract
Here we survey the results on the structure of Fuchsian groups due to Hopf, Hedlund, Sullivan, Nicholls, Pommerenke, and others from the viewpoint of the dynamics of the geodesic and horocycle flows on the corresponding surfaces. Special attention is given to the structure of horocycle orbits; in particular, we construct Fuchsian groups with new types of horocycle orbits which are neither closed nor dense in the nonwandering set. We give a unique classification of Fuchsian groups from the dynamical viewpoint and indicate some open problems.
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This work was partially supported by the Russian Foundation for Fundamental Research and the International Science Foundation.