Higher Edu - Research dev card
Development from the higher education and research community
  • Creation or important update: 24/10/11
  • Minor correction: 25/10/11
Keywords : fundamental domain computation in hyperbolic space for the Bianchi groups

This software was developed (or is under development) within the higher education and research community. Its stability can vary (see fields below) and its working state is not guaranteed.
  • Web site
  • System: UNIX-like
  • Current version: 2-1-0
  • License(s): GPL
  • Status: stable release
  • Support: maintained, no ongoing development
  • Designer(s): Alexander D. Rahm
  • Contact designer(s):
  • Laboratory, service: Fourier Institute


General software features

This program carries out computations of the geometry of a certain class of arithmetic groups (the Bianchi groups), via a proper action on a contractible space. We access their group homology.

In more detail, consider an imaginary quadratic number field Q(√−m), where m is a square-free positive integer. Let A(m) be its ring of integers. The Bianchi groups are the groups SL_2(A(m)).

A wealth of information on the Bianchi groups can be found in monographs of Elstrodt/Grunewald/Mennicke, Benjamin Fine, Maclachlan/Reid, etc. These groups act in a natural way on hyperbolic three-space, which is isomorphic to their associated symmetric space.

The kernel of this action is the centre {±1} of the groups, which makes it useful to study the quotients PSL_2(A(m)) of the groups by their centre, which we again call Bianchi groups.

In 1892, Luigi Bianchi computed fundamental domains for this action for some of these groups. Such a fundamental domain has the shape of a hyperbolic polyhedron (up to some missing vertices), and we call it the Bianchi fundamental polyhedron. The computation of the the Bianchi fundamental polyhedron has been implemented in this program for all Bianchi groups.

The images under SL_2(A(m)) of the faces of this polyhedron provide hyperbolic space with a cellular structure. In order to view clearly the local geometry, we pass to the refined cell complex, which we obtain by subdiving this cell structure until the cell stabilisers in SL_2(A(m)) fix the cells pointwise. We can exploit this cell complex in different ways, in order to see different aspects of the geometry of these groups.

A database with cell complexes computed with is available at

Context in which the software is used

This program is based on the computer algebra system Pari/GP, which is freely available.

The version 2.4.3 (or more recent) of Pari/GP needs to be installed in order to use

Publications related to the software

The homological torsion of SL_2 of the imaginary quadratic integers, Alexander D. Rahm, Accepted for publication at the Transactions of the AMS.

Homologie et K-théorie des groupes de Bianchi, Alexander D. Rahm, Note bilingue anglais/francais, Comptes Rendus Mathématique of the Académie des Sciences - Paris, Volume 349, Issues 11-12, June 2011, Pages 615-619, doi:10.1016/j.crma.2011.05.014 .

The integral homology of PSL_2 of imaginary quadratic integers with nontrivial class group, Alexander D. Rahm and Mathias Fuchs, Journal of Pure and Applied Algebra, DOI: 10.1016/j.jpaa.2010.09.005 .

(Co)homologies et K-théorie de groupes de Bianchi par des modèles géométriques calculatoires, Alexander D. Rahm, Thèse de Doctorat, Université de Grenoble and Universität Göttingen, 2010.
Archived on the server TEL of the CNRS.

On level one cuspidal Bianchi modular forms, Alexander D. Rahm and Mehmet Haluk Sengun, preprint on HAL and arXiv.