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

Let P1 : S1 → S and P2 : S2 → S be non-constant holomorpic maps between closed Riemann surfaces. Associated to the previous data is the fiber product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_{1} \times_{S} S_{2}=\{(x,y) \in S_{1} \times S_{2}: P_{1}(x)=P_{2}(y)\}}$$\end{document} . This is a compact space which, in general, fails to be a Riemann surface at some (possible empty) finite set of points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F \subset S_{1} \times_{S} S_{2}}$$\end{document} . One has that S1 ×SS2 − 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{S_{1}\times_{S} S_{2}}}$$\end{document} . In this paper we prove that the connected components of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{S_{1}\times_{S} S_{2}}}$$\end{document} 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.