In mathematics, a Bianchi group is a group of the form
![{\displaystyle PSL_{2}({\mathcal {O}}_{d})}](//wikimedia.org/api/rest_v1/media/math/render/svg/50a16a046d5b51d34e1620817c8209f4c497c84b)
where d is a positive square-free integer. Here, PSL denotes the projective special linear group and
is the ring of integers of the imaginary quadratic field
.
The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of
, now termed Kleinian groups.
As a subgroup of
, a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space
. The quotient space
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
, was computed by Humbert as follows. Let
be the discriminant of
, and
, the discontinuous action on
, then
![{\displaystyle \operatorname {vol} (\Gamma \backslash \mathbb {H} )={\frac {|D|^{3/2}}{4\pi ^{2}}}\zeta _{\mathbb {Q} ({\sqrt {-d}})}(2)\ .}](//wikimedia.org/api/rest_v1/media/math/render/svg/20986040fcfc70632fa7729cf92fee31b56c3f19)
The set of cusps of
is in bijection with the class group of
. It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.[1]
Maclachlan & Reid (2003) p. 58
- 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.