Bianchi group

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

${\displaystyle PSL_{2}({\mathcal {O}}_{d})}$

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

The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of ${\displaystyle PSL_{2}(\mathbb {C} )}$, now termed Kleinian groups.

As a subgroup of ${\displaystyle PSL_{2}(\mathbb {C} )}$, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$. The quotient space ${\displaystyle M_{d}=PSL_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}}$ 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 ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$, was computed by Humbert as follows. Let ${\displaystyle D}$ be the discriminant of ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$, and ${\displaystyle \Gamma =SL_{2}({\mathcal {O}}_{d})}$, the discontinuous action on ${\displaystyle {\mathcal {H}}}$, then

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

The set of cusps of ${\displaystyle M_{d}}$ is in bijection with the class group of ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$. It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.^[1]

• Bianchi, Luigi (1892). "Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî". Mathematische Annalen. 40 (3). Springer Berlin / Heidelberg: 332–412. doi:10.1007/BF01443558. ISSN 0025-5831. JFM 24.0188.02. S2CID 120341527.
• Elstrodt, Juergen; Grunewald, Fritz; Mennicke, Jens (1998). Groups Acting On Hyperbolic Spaces. Springer Monographs in Mathematics. Springer Verlag. ISBN 3-540-62745-6. Zbl 0888.11001.
• Fine, Benjamin (1989). Algebraic theory of the Bianchi groups. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 129. New York: Marcel Dekker Inc. ISBN 978-0-8247-8192-7. MR 1010229. Zbl 0760.20014.
• Fine, B. (2001) [1994], "Bianchi group", Encyclopedia of Mathematics, EMS Press
• Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Vol. 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.

• Allen Hatcher, Bianchi Orbifolds