In this chapter we collect a number of geometric properties of surfaces which are modelled over the hyperbolic plane. The way of construction of these surfaces is cutting and pasting. For this we shall include surfaces with piece-wise geodesic boundary and prove existence and uniqueness theorems for geodesics in various types of homotopy classes. The chapter ends with an outline of the Fenchel-Nielsen parameters.
KeywordsHomotopy Class Hyperbolic Plane Closed Geodesic Compact Riemann Surface Hyperbolic Surface
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