Bianchi group

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

\( PSL_2(\mathcal{O}_d) \)

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 Bianchi (1892) as a natural class of discrete subgroups of \( PSL_2(\mathbb{C}) \), now termed Kleinian groups.

As a subgroup of \( PSL_2(\mathbb{C}) \), a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space \( \mathbb{H}^3 \). The quotient space \( M_d = PSL_2(\mathcal{O}_d) \backslash\mathbb{H}^3 \) is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi manifold. 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 \( \Gamma=SL_2(\mathcal{O}_d) \), the discontinuous action on \mathcal{H}, then

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

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

References

Maclachlan & Reid (2003) p.58

Bianchi, Luigi (1892). "Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî". Mathematische Annalen (Springer Berlin / Heidelberg) 40: 332–412. doi:10.1007/BF01443558. ISSN 0025-5831. JFM 24.0188.02.
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 129. New York: Marcel Dekker Inc. ISBN 978-0-8247-8192-7. MR 1010229. Zbl 0760.20014.
Fine, B. (2001), "Bianchi group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.

External links

Allen Hatcher, Bianchi Orbifolds

Mathematics Encyclopedia