Quasi-Geodesics and Localization

  • Filippo BracciEmail author
  • Manuel D. Contreras
  • Santiago Díaz-Madrigal
Part of the Springer Monographs in Mathematics book series (SMM)


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.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Filippo Bracci
    • 1
    Email author
  • Manuel D. Contreras
    • 2
  • Santiago Díaz-Madrigal
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Departamento de Matemática Aplicada II and IMUSUniversidad de SevillaSevillaSpain

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