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

The fiber product of Riemann surfaces: a Kleinian group point of view 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

, Volume 1 (1) – Feb 3, 2011
9 pages

/lp/springer-journals/the-fiber-product-of-riemann-surfaces-a-kleinian-group-point-of-view-Gk6GrErsWd
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-010-0003-9
Publisher site
See Article on Publisher Site

### 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 3, 2011