In mathematics, a Bianchi group is a group of the form
P S L 2 ( O d ) where d is a positive square-free integer. Here, PSL denotes the projective special linear group and O d is the ring of integers of the imaginary quadratic field Q ( − d ) .
The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of P S L 2 ( C ) , now termed Kleinian groups.
As a subgroup of P S L 2 ( C ) , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H 3 . The quotient space M d = P S L 2 ( O d ) ∖ H 3 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 Q ( − d ) , was computed by Humbert as follows. Let D be the discriminant of Q ( − d ) , and Γ = S L 2 ( O d ) , the discontinuous action on H , then
v o l ( Γ ∖ H ) = | D | 3 2 4 π 2 ζ Q ( − d ) ( 2 ) . The set of cusps of M d is in bijection with the class group of Q ( − d ) . It is well known that any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.
