Bianchi group

In mathematics, a Bianchi group is a group of the form $${\displaystyle {\text{PSL}}_{2}({\mathcal {O}}_{d})}$$ where d is a positive square-free integer. Here, ${\displaystyle {\text{PSL}}_{2}}$ 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 ${\displaystyle {\text{PSL}}_{2}(\mathbb {C} )}$, now termed Kleinian groups.

Geometry

As a subgroup of ${\displaystyle {\text{PSL}}_{2}(\mathbb {C} )}$, a Bianchi group acts by orientation-preserving isometries on three-dimensional hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$. The quotient $${\displaystyle M_{d}={\text{PSL}}_{2}({\mathcal {O}}_{d})\backslash \mathbb {H} ^{3}}$$ is a non-compact, hyperbolic 3-fold of finite volume, called a Bianchi orbifold. An exact formula for its volume in terms of the Dedekind zeta function of the underlying imaginary quadratic field was computed by Humbert. If ${\displaystyle D}$ is the discriminant of ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$, then $${\displaystyle \operatorname {vol} (M_{d})={\frac {|D|^{3/2}}{4\pi ^{2}}}\zeta _{\mathbb {Q} ({\sqrt {-d}})}(2).}$$

The cusps of ${\displaystyle M_{d}}$ are in bijection with the ideal classes of ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$. Consequently, the number of cusps is equal to the class number of the field.

Arithmetic properties

Bianchi groups are arithmetic Kleinian groups. Every non-cocompact arithmetic Kleinian group is commensurable, up to conjugacy in ${\displaystyle {\text{PSL}}_{2}(\mathbb {C} )}$, with a Bianchi group.

References

• 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.

External links

• Allen Hatcher, Bianchi Orbifolds

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