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