Bianchi group

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


 * $$PSL_2(\mathcal{O}_d)$$

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

The groups were first studied by as a natural class of discrete subgroups of $$PSL_2(\mathbb{C})$$, now termed Kleinian groups.

As a subgroup of $$PSL_2(\mathbb{C})$$, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space $$\mathbb{H}^3$$. The quotient space $$M_d = PSL_2(\mathcal{O}_d) \backslash\mathbb{H}^3$$ 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 $$\mathbb{Q}(\sqrt{-d})$$, was computed by Humbert as follows. Let $$D$$ be the discriminant of $$\mathbb{Q}(\sqrt{-d})$$, and $$\Gamma=SL_2(\mathcal{O}_d)$$, the discontinuous action on $$\mathcal{H}$$, then


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

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