Paths and Geodesics

Abstract

With the Killing-Hopf theorem, that every geometric surface is of the form S²/Γ, ℝ²/Γ, or ℍ²/Γ, the problem of classifying surfaces is replaced by the problem of classifying groups Γ. In the spherical and euclidean cases this problem is easy to solve, as we have seen in Chapters 2 and 3, because there are only a small number of possibilities. However, in the hyperbolic case the number of possibilities is infinite, and the problem is best clarified by taking a new viewpoint, halfway between geometry and group theory. This is the viewpoint of topology briefly mentioned in Section 5.6.