On the Connectivity of the Branch Locus of the Schottky Space

Let M be a handlebody of genus g ≥ 2. The space T (M), that parametrizes marked Kleinian structures on M up to isomorphisms, can be identified with the space MSg 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 Sg, the Schottky space of rank g, and it carries the structure of a complex orbifold. In these identifications, the projection map π : T (M) → M(M) corresponds to the map from MSg onto Sg 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 ≥ 4 even, and (iii) M(M) is connected for g ≥ 3 odd.