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 \(\bar S{}_ g\).
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) |
