Quasi-Geodesics and Localization

Abstract

In this chapter we show that hyperbolic neighborhoods of geodesic rays correspond to Stolz regions and thus geodesics can be used to understand non-tangential and orthogonal behavior of pre-images under Riemann maps of sequences or curves converging to the boundary in simple connected domains. As it is essentially impossible to detect geodesics in simply connected domains, we introduce the notion of Gromov’s quasi-geodesics, which are usually much simpler to find, and prove the so-called Shadowing Lemma, which states that close to every quasi-geodesic there is a geodesic.