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The extended Teichmüller space. (English) Zbl 0662.32021
A partial compactification \(\bar T_ g\) of the Teichmüller space \(T_ g\) of Riemann surfaces of genus g is constructed and given a structure as complexed ringed space such that the Teichmüller modular group \(G_ g\) acts discontinuously on \(\bar T_ g\), and the orbit space \(\bar T_ g/G_ g\) is as a complex space isomorphic to the moduli \(\bar M_ g\) of stable Riemann surfaces of genus g. The standard family of Riemann surfaces over \(T_ g\) is extended to a family of stable Riemann surfaces over \(\bar T_ g\), and its universal property is stated.
The points in \(\bar T_ g\) are stable Riemann surfaces together with a “standard” set of generators, for the fundamental group. The topology on \(\bar T_ g\) is inherited from the closely related augmented Teichmüller space of Bers and W. Abikoff [Ann. Math., II. Ser. 105, 29-44 (1977; Zbl 0347.32010)]. The structure of complex ringed space is derived as follows: for any x there exist neighbourhoods U of x invariant under the stabilizer \(G_ x\) of x in \(G_ g\), and \(U/G_ x\) is an unramified covering of an open domain in the extended Schottky space, a complex manifold introduced in a joint paper of L. Gerritzen and the author [J. Reine Angew. Math. 389, 190-208 (1988; Zbl 0639.30040)]. Then locally around x, \(\bar T_ g\) is given the complex structure of \(U/G_ x.\)
In the final section of the paper an algebraic property of the Teichmüller modular group is conjectured and its consequences for the topological structure of \(\bar T_ g\) are discussed.
Reviewer: F.Herrlich

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
32J05 Compactification of analytic spaces
30F99 Riemann surfaces
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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References:
[1] Abikoff, W.: Degenerating families of Riemann surfaces. Ann. Math.105, 29–44 (1977) · Zbl 0347.32010 · doi:10.2307/1971024
[2] Birman, J.: Braids, links and mapping class groups. Ann. Math. Studies no. 82 (1974)
[3] Casson, A., Long, D., Algorithmic compression of surface automorphisms. Invent. Math.81, 295–303 (1985) · Zbl 0589.57008 · doi:10.1007/BF01389054
[4] Gerritzen, L., Herrlich, F.: The extended Schottky spaces. J. Reine Angew. Math.389, 190–208 (1988) · Zbl 0639.30040
[5] Harvey, W.J.: Spaces of Discrete Groups. In: Discrete groups and automorphic functions. Proc. Lond Math. Soc. Instructional Conference, London 1977 · Zbl 0411.30033
[6] Hejhal, D.: On Schottky and Teichmüller Spaces. Adv. Math.15, 133–156 (1975) · Zbl 0307.32018 · doi:10.1016/0001-8708(75)90128-0
[7] Marden, A.: Geometric complex coordinates for Teichmüller space. In: Yau, S. T. (ed.), Mathematical aspects of String theory Adv. Series in Math, Phys.1, 341–354 (1987) · Zbl 0678.32014
[8] Powell, J.: Homeomorphisms ofS 3 leaving a Heegard surface invariant. Trans. Am. Math. Soc.257, 193–216 (1980) · Zbl 0445.57008
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