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 Bianchi (1892) as a natural class of discrete subgroups of

, now termed Kleinian groups.

, 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.

, 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.

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