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Mathematical 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 O d {\displaystyle {\mathcal {O}}_{d}} is the ring of integers of the imaginary quadratic field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} .
The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of PSL 2 ( C ) {\displaystyle {\text{PSL}}_{2}(\mathbb {C} )} , now termed Kleinian groups.
As a subgroup of PSL 2 ( C ) {\displaystyle {\text{PSL}}_{2}(\mathbb {C} )} , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H 3 {\displaystyle \mathbb {H} ^{3}} . The quotient space M d = PSL 2 ( O d ) ∖ H 3 {\displaystyle M_{d}={\text{PSL}}_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}} 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 Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , was computed by Humbert as follows. Let D {\displaystyle D} be the discriminant of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , and Γ = SL 2 ( O d ) {\displaystyle \Gamma ={\text{SL}}_{2}({\mathcal {O}}_{d})} , the discontinuous action on H {\displaystyle {\mathcal {H}}} , then
The set of cusps of M d {\displaystyle M_{d}} is in bijection with the class group of Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} . It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.[1]
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