Punctured torus groups and 2-bridge knot groups. I
In the late 70's \textit{T. Jørgensen} has made a series of detailed studies on the space \(\mathcal{QF}\) of quasifuchsian (once) punctured torus groups from the view point of their Ford fundamental domains. These studies are summarized in [Lond. Math. Soc. Lect. Note Ser. 299, 183--207 (2003; Zbl 1061.30038)]. He gave a complete description of the combinatorial structure of the Ford domain of every quasifuchsian punctured torus group, and showed that the space \(\mathcal{QF}\) can be described in terms of the Farey triangulations, or the modular diagrams. The present monograph is Part 1 of a book intended to give a full account of Jørgensen's theory of punctured torus Kleinian groups and its generalization. The authors give an intuitive and self-contained description of the theory with a complete proof. The method is based on Poincaré's theorem on fundamental polyhedra. This geometric approach enables them to extend Jørgensen's theory beyond quasifuchsian spaces and to apply it to knot theory. They find a very natural way to understand the hyperbolic structure and the canonical decompositions of the 2-bridge link complements in the context of Jørgensen's work. This monograph written by well-known experts is an excellent presentation of Jørgensen's theory with many informative illustrations. It will be very useful for researches working on modern problems in quasifuchsian group theory, hyperbolic and geometry and hyperbolic knot theory.
scientific article; zbMATH DE number 5152430