Geodesics on quotient manifolds and their corresponding limit points
Let \(\Gamma\) be a Fuchsian or Kleinian group acting on the unit circle or ball \(B\); let \(S\) be the unit circle or sphere, i.e., the boundary of \(B\). Let \(L\) be the limit set of \(\Gamma\). The author considers half-geodesics, i.e., maps \(\gamma:[0,\infty[ \rightarrow B\) which are geodesic. This defines a point \(\xi(\gamma)\in S\). Let \(\phi_\gamma(t)\) be the distance, in \(\Gamma \backslash B\) between \(\gamma(0)\) and \(\gamma(t)\). If \(\phi_\gamma\) is bounded then \(\xi(\gamma)\) lies in the conical limit set of \(\Gamma\). The main theorem of this paper describes the geometrical nature of the limit point \(\xi(\gamma)\) in terms of the asymptotics \(\phi_\gamma\). The proof uses the methods of hyperbolic geometry. The paper concludes with some related results and examples.
scientific article; zbMATH DE number 1997313
scientific article