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 orbifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field, was computed by Humbert as follows. Let D be the discriminant of, and , the discontinuous action on \mathcal{H}, then The set of cusps of M_d is in bijection with the class group of. It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.