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On the Maskit coordinates of Teichmüller spaces and modular transformations. (English) Zbl 0698.32013

Let S be a finite Riemann surface of type (p,n) with \(3p-3+n>0.\) In this case the Teichmüller space T(S) of S is a \(3p-3+n\) dimensional complex manifold. B. Maskit [Bull. Am. Math. Soc. 80, 773-777 (1974; Zbl 0292.30016)] has observed that there is some canonical embedding \(T(S)\to U^{3p-3+n}\), where U is the upper half plane. The embedding yield global coordinates of T(S) called the Maskit coordinates. In the first part of the paper the authors obtain in terms of Maskit coordinates the canonical form of modular transformation induced by completely reduced quasiconformal mappings of S.
The second part of the paper deals with the problem: what kind of modular transformations leave some Teichmüller disc invariant? It is known that each hyperbolic transformation has an invariant Teichmüller disc [L. Bers, Acta Math. 141, 73-98 (1978; Zbl 0389.30018)] and each pseudo- hyperbolic transformation has none. With respect to parabolic transformations it is also known [A. Marden and H. Masur, Math. Scand. 36, 211-228 (1975; Zbl 0318.32019)] that a product of Dehn twists about an admissible set curves leaves Teichmüller disc invariant if the factors of the product have simultaneously positive orders or negative orders. The converse of this fact, which was suggested early by Earle, is shown in this paper.
Reviewer: A.D.Mednych

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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