Bianchi group

Short description: Mathematical group

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

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, PSL denotes the projective special linear group and ${{𝒪}}_d$ is the ring of integers of the imaginary quadratic field $\mathbb{\mathbb{Q}}(\sqrt{-d})$.

The groups were first studied by Bianchi (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 $M_d={\text{PSL}}_2({{𝒪}}_d)\mathrm{\setminus }{\mathbb{ℍ}}^3$ 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 $\mathbb{\mathbb{Q}}(\sqrt{-d})$, was computed by Humbert as follows. Let $D$ be the discriminant of $\mathbb{\mathbb{Q}}(\sqrt{-d})$, and $\mathrm{\Gamma }={\text{SL}}_2({{𝒪}}_d)$, the discontinuous action on ${\mathscr{H}}$, then

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

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

References

• Bianchi, Luigi (1892). "Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî". Mathematische Annalen (Springer Berlin / Heidelberg) 40 (3): 332–412. doi:10.1007/BF01443558. ISSN 0025-5831. https://zenodo.org/record/2260508. 
• Elstrodt, Juergen; Grunewald, Fritz; Mennicke, Jens (1998). Groups Acting On Hyperbolic Spaces. Springer Monographs in Mathematics. Springer Verlag. ISBN 3-540-62745-6. 
• Fine, Benjamin (1989). Algebraic theory of the Bianchi groups. Monographs and Textbooks in Pure and Applied Mathematics. 129. New York: Marcel Dekker Inc.. ISBN 978-0-8247-8192-7. https://books.google.com/books?id=1D6crOEoRFEC. 
• Hazewinkel, Michiel, ed. (2001), "Bianchi group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Bianchi_group 
• Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. 219. Springer-Verlag. ISBN 0-387-98386-4. 

External links

• Allen Hatcher, Bianchi Orbifolds