Bianchi group
In mathematics, a Bianchi group is a group of the form
where d is a positive square-free integer. Here, PSL denotes the projective special linear group and is the ring of integers of the imaginary quadratic field.
The groups were first studied by as a natural class of discrete subgroups of, now termed Kleinian groups.
As a subgroup of, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space. The quotient space is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi manifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field, was computed by Humbert as follows. Let be the discriminant of, and, the discontinuous action on, then
The set of cusps of is in bijection with the class group of. It is well known that
any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.
