Bianchi group

Short description: Mathematical group

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

[math]\displaystyle{ PSL_2(\mathcal{O}_d) }[/math]

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

The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of [math]\displaystyle{ PSL_2(\mathbb{C}) }[/math], now termed Kleinian groups.

As a subgroup of [math]\displaystyle{ PSL_2(\mathbb{C}) }[/math], a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space [math]\displaystyle{ \mathbb{H}^3 }[/math]. The quotient space [math]\displaystyle{ M_d = PSL_2(\mathcal{O}_d) \backslash\mathbb{H}^3 }[/math] 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 [math]\displaystyle{ \mathbb{Q}(\sqrt{-d}) }[/math], was computed by Humbert as follows. Let [math]\displaystyle{ D }[/math] be the discriminant of [math]\displaystyle{ \mathbb{Q}(\sqrt{-d}) }[/math], and [math]\displaystyle{ \Gamma=SL_2(\mathcal{O}_d) }[/math], the discontinuous action on [math]\displaystyle{ \mathcal{H} }[/math], then

[math]\displaystyle{ \operatorname{vol}(\Gamma\backslash\mathbb{H})=\frac{|D|^{3/2}}{4\pi^2}\zeta_{\mathbb{Q}(\sqrt{-d})}(2) \ . }[/math]

The set of cusps of [math]\displaystyle{ M_d }[/math] is in bijection with the class group of [math]\displaystyle{ \mathbb{Q}(\sqrt{-d}) }[/math]. 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