Bianchi group

{{For|the 3-dimensional Lie groups or Lie algebras|Bianchi classification}}

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

: $\text{PSL}_2(\mathcal{O}_d)$

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 {{harvs|txt|last=Bianchi|authorlink=Luigi Bianchi|year=1892}} as a natural class of discrete subgroups of $\text{PSL}_2(\mathbb{C})$ , now termed Kleinian groups.

As a subgroup of $\text{PSL}_2(\mathbb{C})$ , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space $\mathbb{H}^3$ . The quotient space $M_d = \text{PSL}_2(\mathcal{O}_d) \backslash\mathbb{H}^3$ 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 $\Gamma=\text{SL}_2(\mathcal{O}_d)$ , 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.Maclachlan & Reid (2003) p. 58

References

{{cite journal | last1=Bianchi | first1=Luigi | title=Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01443558 | year=1892 | journal=Mathematische Annalen | issn=0025-5831 | volume=40 | issue=3 | jfm=24.0188.02 | pages=332–412 | s2cid=120341527 | url=https://zenodo.org/record/2260508 }}
{{cite book | first1=Juergen | last1=Elstrodt | first2=Fritz | last2=Grunewald | first3=Jens | last3=Mennicke | title=Groups Acting On Hyperbolic Spaces | series=Springer Monographs in Mathematics | publisher=Springer Verlag | year=1998 | isbn=3-540-62745-6 | zbl=0888.11001 }}
{{cite book | last1=Fine | first1=Benjamin | title=Algebraic theory of the Bianchi groups | url=https://books.google.com/books?id=1D6crOEoRFEC | publisher=Marcel Dekker Inc. | location=New York | series=Monographs and Textbooks in Pure and Applied Mathematics | isbn=978-0-8247-8192-7 | mr=1010229 | year=1989 | volume=129 | zbl=0760.20014 }}
{{springer|id=Bianchi_group|first=B.|last=Fine|title=Bianchi group}}
{{cite book | first1=Colin | last1=Maclachlan | first2=Alan W. | last2=Reid | title=The Arithmetic of Hyperbolic 3-Manifolds | series=Graduate Texts in Mathematics | publisher=Springer-Verlag | volume=219 | year=2003 | isbn=0-387-98386-4 | zbl=1025.57001 }}

External links

Allen Hatcher, [https://pi.math.cornell.edu/~hatcher/bianchi.html Bianchi Orbifolds]

Category:Group theory