Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 635-654

ON THE CONNECTIVITY OF THE BRANCH LOCUS OF THE SCHOTTKY SPACE

Rubén A. Hidalgo and Milagros Izquierdo

Universidad Técnica Federico Santa María, Departamento de Matemática
Casilla 110-V Valparaiso, Chile; ruben.hidalgo 'at' usm.cl

Linköpings Universitet, Mathematiska Institutionen
581 83 Linköping, Sweden; milagros.izquierdo 'at' liu.se

Abstract. Let M be a handlebody of genus g \geq 2. The space T(M), that parametrizes marked Kleinian structures on M up to isomorphisms, can be identified with the space MS_g of marked Schottky groups of rank g, so it carries a structure of complex manifold of finite dimension 3(g - 1). The space M(M) parametrizing Kleinian structures on M up to isomorphisms, can be identified with S_g, the Schottky space of rank g, and it carries the structure of a complex orbifold. In these identifications, the projection map π : T(M) \to M(M) corresponds to the map from MS_g onto S_g that forgets the marking. In this paper we observe that the singular locus B(M) of M(M), that is, the branch locus of π, has (i) exactly two connected components for g = 2, (ii) at most two connected components for g \geq 4 even, and (iii) M(M) is connected for g \geq 3 odd.

2010 Mathematics Subject Classification: Primary 30F10, 30F40.

Key words: Moduli space, branch locus, Schottky space, Schottky group, handlebody, Riemann surface, Kleinian group, Fuchsian group.

Reference to this article: R.A. Hidalgo and M. Izquierdo: On the connectivity of the branch locus of the Schottky space. Ann. Acad. Sci. Fenn. Math. 39 (2014), 635-654.

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doi:10.5186/aasfm.2014.3942