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.