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 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  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 every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group. References

External links
 Allen Hatcher, Bianchi Orbifolds

Group theory