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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:
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