computes a fundamental domain for the Bianchi groups in hyperbolic 3-space, the associated quotient space and essential information about the group homology and equivariant K-homology of the Bianchi groups.

Fundamental polyhedron for the Bianchi group of discriminant -427, as computed with and visualized with geomview.

Using the fundamental domains and the orbifold which is computed for them, the dimensions of spaces of Bianchi modular forms can be determined.

The program has been validated in the PLUME index of the CNRS.
The intensive computations award C3I has been attributed to A. Rahm's PhD thesis, which was the framework of
This program is based on the specialized computer algebra system Pari/GP; and version 2.4.3 or higher of the latter is required to execute
Pari/GP is available free of charge on the Pari/GP development headquarters website. is part of the GP scripts library of Pari/GP Development Center.

A user surface for in SAGE has been created by Atin Modi. It has been made compatible with year 2020 versions of sagemath (based on Python 3) by Joel Sjogren. If you have an installation of SAGE on your hard disk, then you can download BianchiGPinSage.tar.gz, unpack it (e.g. with the command "tar -xzvf"), and type the command
after having started SAGE from the unpacked directory.
The old versions (based on Python 2, and no more operational in current versions of SAGE) are here.

The current stable release (version 2.1.7) of is downloadable from here free of charge, covered by the GNU General Public License .
A description of is given in the appendix of the PhD thesis of Alexander D. Rahm.

See also the cell complexes database computed with

Luigi Bianchi (1856-1928) contains the following libraries of functions and procedures.

In old Pari/GP versions (2.4.3 to 2.5.x), you can also use version 2.1.3, which has been stable for a long time - but please do not use it in more recent versions of Pari/GP.

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