The fiber product of Riemann surfaces: a Kleinian group point of view

Abstract

Let P 1 : S 1 → S and P 2 : S 2 → S be non-constant holomorpic maps between closed Riemann surfaces. Associated to the previous data is the fiber product \({S_{1} \times_{S} S_{2}=\{(x,y) \in S_{1} \times S_{2}: P_{1}(x)=P_{2}(y)\}}\) . This is a compact space which, in general, fails to be a Riemann surface at some (possible empty) finite set of points \({F \subset S_{1} \times_{S} S_{2}}\) . One has that S 1 × S S 2 − F is a finite collection of analytically finite Riemann surfaces. By filling out all the punctures of these analytically finite Riemann surfaces, we obtain a finite collection of closed Riemann surfaces; whose disjoint union is the normal fiber product \({\widetilde{S_{1}\times_{S} S_{2}}}\) . In this paper we prove that the connected components of \({\widetilde{S_{1}\times_{S} S_{2}}}\) of lowest genus are conformally equivalents if they have genus different from one (isogenous if the genus is one). A description of these lowest genus components are provided in terms of certain class of Kleinian groups; B-groups.