Dirichlet Points , Garnett Points , Andinfinite Ends of Hyperbolic Surfaces

The end of a hyperbolic surface is studied in terms of the behavior at innnity of geodesics on the surface. For a class of surfaces called untwisted utes it is possible to give a fairly precise description of the ending geometry. From the point of view of a Fuchsian group representing such a surface this provides new information about the existence of Dirichlet and Garnett points. 0. Introduction In the study of surfaces of nite topological type the analytic and geometric viewpoints have long been inextricably interwoven. This has been less true in the study of surfaces of innnite type, where even geometrically oriented thinking has tended to focus primarily on generic behavior. Our goal in this paper is to employ hyperbolic geometric techniques to investigate an innnite end of a hyperbolic manifold and especially a hyperbolic surface. We shall be interested in the behavior of speciic classes of geodesics which carry information about the geometry of the end. Such classes of geodesics can have measure zero, or even be nite, and consequently this type of behavior is invisible to the usual analytic approaches. Let M be a hyperbolic manifold and let : 0; 1) ! M be a geodesic ray, which we assumed to be parameterized by arc length. Deene the function (t) = t ? d M ? (0); (t) , where d M denotes distance as deened by the hyper-bolic metric. The ray is then said to be horocyclic, critical, or subcritical if (t) is respectively, unbounded, zero, or nonzero but bounded. A critical ray could be said to travel directly out a non-compact end of M , and a subcritical ray to travel almost directly out an end. Critical and subcritical rays are closely related to the more familiar Dirichlet and Garnett points associated with Fuchsian and Kleinian groups. Viewed as asymptotic classes of geodesic rays one begins to see exactly how their existence and their particular travels depend on the geometry of the hyperbolic manifold on which they reside. Most of our attention will be focused on the untwisted ute surfaces studied by Basmajian 1]. A ute is a complete hyperbolic surface which is homeomorphic to the innnite cylinder K = S 1 (0; 1) with the set of points f(1; n) j n 2 Ng