Bianchi group

In mathematics, a Bianchi group is a group of the form

${\displaystyle PSL_{2}({\mathcal {O}}_{d})}$

where d is a positive square-free integer. Here, PSL denotes the projective special linear group and ${\displaystyle {\mathcal {O}}_{d}}$ is the ring of integers of the imaginary quadratic field ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$.

The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of ${\displaystyle PSL_{2}(\mathbb {C} )}$, now termed Kleinian groups.

As a subgroup of ${\displaystyle PSL_{2}(\mathbb {C} )}$, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$. The quotient space ${\displaystyle M_{d}=PSL_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}}$ 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 ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$, was computed by Humbert as follows. Let ${\displaystyle D}$ be the discriminant of ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$, and ${\displaystyle \Gamma =SL_{2}({\mathcal {O}}_{d})}$, the discontinuous action on ${\displaystyle {\mathcal {H}}}$, then

${\displaystyle \operatorname {vol} (\Gamma \backslash \mathbb {H} )={\frac {|D|^{3/2}}{4\pi ^{2}}}\zeta _{\mathbb {Q} ({\sqrt {-d}})}(2)\ .}$

The set of cusps of ${\displaystyle M_{d}}$ is in bijection with the class group of ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$. It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.^[1]