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 Bianchi.gp is available at

http://www.wisdom.weizmann.ac.il/~rahm/cellComplex...