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

作者：Hidalgo Ruben A^*

来源：Analysis and Mathematical Physics, 2011, 1(1): 37-45.

DOI：10.1007/s13324-010-0003-9

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 x(S) S-2 = {(x, y) is an element of S-1 x 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 of S-1 x(S) S-2. One has that S-1 x(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 (S-1 x(S) S-2) over tilde. In this paper we prove that the connected components of (S-1 x(S) S-2) over tilde 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.

• 出版日期2011-3