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The extended Schottky space. (English) Zbl 0639.30040
The extended Schottky space $$\bar S{}_ g$$ is a fine moduli space for stable Riemann surfaces X of genus $$g\geq 2$$ with Schottky structure. A Schottky structure on X is essentially given by a Schottky covering $$(=$$ uniformization) of X. The canonical mapping $$\mu$$ of $$\bar S{}_ g$$ into the moduli space $$\bar M{}_ g$$ of stable Riemann surfaces is surjective and forgets the Schottky structure; its fibres are discrete. There is a canonical action of Out $$\Gamma$$, $$\Gamma$$ free group of rank g, on $$\bar S{}_ g$$ and $$\mu$$ : $$\bar S{}_ g\to \bar M_ g$$ factors over $$\bar S{}_ g/Out \Gamma$$. The induced map $${\bar \mu}$$: $$\bar S{}_ g/Out \Gamma \to \bar M_ g$$ is bianalytic in a vicinity of the subset of totally degenerate Riemann surfaces.
One can use the Schottky space $$\bar S{}_ g$$ to construct the partial compactification $$\bar T{}_ g$$ of the Teichmüller space $$T_ g$$ such that $$\bar T{}_ g$$ modulo the Teichmüller modular group gives $$\bar M{}_ g.$$
This shows that one can quite well investigate the moduli space $$\bar M{}_ g$$ through $$\bar S{}_ g$$; thus providing an alternative to the usual method by multi-canonical embeddings, Hilbert schemes and geometric quotients by $$PGL_ n$$ as given by D. Mumford and D. Gieseker, see D. Gieseker, Lectures on moduli of curves (1982; Zbl 0534.14012). The idea to extend the space $$S_ g$$ of conjugacy classes of Schottky subgroups of $$PGL_ 2({\mathbb{C}})$$ in order to describe the boundary of $$M_ g$$ is due to L. Bers [see Automorphic forms for Schottky groups, Adv. Math. 16, 332-361 (1975; Zbl 0327.32011)].
The main idea in the paper consists in using the cross-ratios of the fixed points of all elements of a group $$\Gamma$$ acting on a tree of Riemann spheres as coordinates for an infinite dimensional space B on which $$\Gamma$$ acts. The invariant subspace $$B^{\Gamma}$$ turns out to classify $$\Gamma$$-actions on trees of Riemann spheres, and $$\bar S{}_ g$$, defined as the space of Schottky actions of $$\Gamma$$, is an open subspace of $$B^{\Gamma}$$. It is shown that $$\bar S{}_ g$$ is a complex space of dimension 3g-3 and that the usual Schottky space $$S_ g$$ is an open dense subspace of $$\bar S{}_ g$$.
Reviewer: L.Gerritzen

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
##### Keywords:
Schottky covering; Schottky space
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