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Concept
Bianchi group
Summary
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.
Official source
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