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Extending Schottky space: Parabolic completion vs. Teichmüller space of hyperbolic handlebodies. (English) Zbl 0849.30035

Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik. Heft 2. Bochum: Math. Inst., Univ. Bochum, ix, 115 p. (1993).
The thesis consist of three chapters.
Chapter I: Prolongena contains the basic definition and results of handlebody topology and of the theory of Schottky groups. In particular, an algebraic geometry treatment of the notion of moduli space is given there.
Chapter II: The parabolic completion is devoted to the study of a certain type of degeneration of Schottky groups. The analysis is based on the combinatorics of system of \(X_g\)-trivial Jordan curves on \(F_g\), the topological model of closed Riemann surface of genus \(g\).
The used combinatorial concept of such curve system is the simplicial complex of curves introduced by Harvey. The relation of this curve complex to another combinatorial object, based on the investigation of cut systems is clarified.
The limit set of handlebody group on the Thurston sphere is investigated. The realization of both of the curve system and of Teichmüller spaces of stable Riemann surfaces in this limit set is given.
The construction of the parabolic completion of Schottky space is given in which this space turns out to be homeomorphic to the Gerritzen-Herrlich extension of Schottky space.
Chapter III: Hyperbolic handlebodies is devoted to the study of Schottky groups forming a special class of those Kleinian groups which induce a hyperbolic structure on the interior of a handlebody. The totality of all these structures is collected in a space \(T(X_g)\) which is constructed in analogy to the Teichmüller space of closed hyperbolic surfaces. A topology on \(T(X_g)\) is introduced by considering quasi-isometries in the sense of Gromov on exhausting sequences of compacta in \((X_g)^0\). The characterizing of the uniformizing groups of these handlebody structures as geometrically tame, free Kleinian groups is given.
It is shown that the boundary of convex core of the associated Kleinian manifold for non-Fuchsian Schottky groups turns out to be a pleated surface in the sense of Thurston. This pleated surface gives two kinds of data: the hyperbolic structure of the surface and a intrinsically defined measured lamination. It is proven that non-Fuchsian Schottky groups can be reconstructed from the knowledge of these data. An interesting numerical invariant for the Fuchsian Schottky group is found and used to estimate the number of connected components of the Fuchsian locus in the Schottky space.
The notion of characteristic lamination for Fuchsian Schottky groups is introduced and proved that assigning to a purely hyperbolic, geometrically tame group its characteristic lamination defines a continuous map from a purely hyperbolic locus of \(T(x_g)\) into the lamination space whose construction is based on the Masur domain of the handlebody group.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable

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