# Bianchi group

For the 3-dimensional Lie groups or Lie algebras, see Bianchi classification.

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 Bianchi (1892) 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. An exact formula for the volume, in terms of the Dedekind zeta function of the base field $\mathbb{Q}(\sqrt{-d})$, was computed by Humbertas 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

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

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

## References

1. ^ Maclachlan & Reid (2003) p.58