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 formula_2 is the ring of integers of the imaginary quadratic field formula_3.
The groups were first studied by as a natural class of discrete subgroups of formula_4, now termed Kleinian groups.
As a subgroup of formula_4, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space formula_6. The quotient space formula_7 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 formula_3, was computed by Humbert as follows. Let formula_9 be the discriminant of formula_3, and formula_11, the discontinuous action on formula_12, then
The set of cusps of formula_14 is in bijection with the class group of formula_3. It is well known that 
any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.