Bianchi group

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

$PS{L}_2({𝒪}_d)$

where d is a positive square-free integer. Here, PSL 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 $PS{L}_2(ℂ)$, now termed Kleinian groups.

As a subgroup of $PS{L}_2(ℂ)$, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space ${ℍ}^3$. The quotient space $M_d=PS{L}_2({𝒪}_d)\mathrm{\setminus }{ℍ}^3$ 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 $\mathrm{\Gamma }=S{L}_2({𝒪}_d)$, the discontinuous action on $\mathscr{H}$, then

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

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.

References

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External links

• Allen Hatcher, Bianchi Orbifolds

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