Bianchi group

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

${\text{PSL}}_2({𝒪}_d)$

where d is a positive square-free integer. Here, ${\text{PSL}}_2$ denotes the projective special linear group and ${𝒪}_d$ is the ring of integers of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$.

The groups were first studied by Template:Harvs as a natural class of discrete subgroups of ${\text{PSL}}_2(ℂ)$, now termed Kleinian groups.

Geometry

As a subgroup of ${\text{PSL}}_2(ℂ)$, a Bianchi group acts by orientation-preserving isometries on three-dimensional hyperbolic space ${ℍ}^3$. The quotient

$M_d={\text{PSL}}_2({𝒪}_d)\mathrm{\setminus }{ℍ}^3$

is a non-compact, hyperbolic 3-fold of finite volume, called a Bianchi orbifold. An exact formula for its volume in terms of the Dedekind zeta function of the underlying imaginary quadratic field was computed by Humbert. If $D$ is the discriminant of $\mathbb{Q}(\sqrt{-d})$, then

$\mathrm{vol}(M_d)=\frac{|D{|}^{3/2}}{4{\pi }^2}{\zeta }_{\mathbb{Q}(\sqrt{-d})}(2).$

The cusps of $M_d$ are in bijection with the ideal classes of $\mathbb{Q}(\sqrt{-d})$. Consequently, the number of cusps is equal to the class number of the field.

Arithmetic properties

Bianchi groups are arithmetic Kleinian groups. Every non-cocompact arithmetic Kleinian group is commensurable, up to conjugacy in ${\text{PSL}}_2(ℂ)$, with a Bianchi group.

References

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External links

• Allen Hatcher, Bianchi Orbifolds

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