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For the 3-dimensional Lie groups or Lie algebras, see Bianchi classification.

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


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]


  1. ^ Maclachlan & Reid (2003) p.58

External links[edit]

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Bianchi_group — Please support Wikipedia.
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89 news items

Cycling Weekly

Cycling Weekly
Tue, 24 Nov 2015 10:05:16 -0800

Stephan Gubler has a growing collection of Bianchi bikes. The current number is 17 but he is already looking for more - the classic N+1. Bianchi group shot. We all like admiring beautiful bikes, almost as much as we like riding them. The fabled N+1 ...

MIT Technology Review

MIT Technology Review
Tue, 15 Dec 2015 21:10:41 -0800

From those 1,300 drugs and chemicals, Bianchi's group narrowed the list to 10. In her lab, the compounds are mixed into food and fed to mice, including pregnant females carrying pups, about half of whom have a condition that mimics Down syndrome.

Bike Europe

Bike Europe
Thu, 24 Dec 2015 00:15:00 -0800

BERGAMO, Italy – Cycleurope owned Bianchi partners together with Ducati Motor Holding to manufacture a new range of Ducati bicycles, including junior, MTB, sport and city models. Ducati Motor Holding has already been on the bicycle market for several ...

Bike Biz

Bike Biz
Tue, 22 Dec 2015 01:53:34 -0800

Bianchi has partnered with Ducati Motor Holding to manufacture a new range of bicycles spanning kids through to sport road and MTB. The Italian duo will also link on distribution, with bicycles to be supplied to independent retailers via Alpen, a ...

MedicalResearch.com (blog)

MedicalResearch.com (blog)
Wed, 14 Oct 2015 15:06:47 -0700

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La Provincia di Como

La Provincia di Como
Wed, 08 Oct 2014 06:18:45 -0700

Al Centro sportivo di Casnate con Bernate si terrà il “Bianchi Group Trophy – Memorial Chicco Marino”. Il torneo internazionale di tennis in carrozzina è promosso dall'Osha Asp Como e dal Tennis Club Lariano con il sostegno del Comitato regionale ...


Wed, 20 May 2015 03:57:28 -0700

The other partners in the consortium are Vienna International Airport with 53.24%, and the Bianchi Group, with 10.63%. De Gabriele insisted that the company has no say in sale of shares, adding “it's completely up to direct or indirect shareholders ...


Fri, 10 Apr 2015 00:52:02 -0700

Canadian engineering giant SNC-Lavalin are 36.1% shareholders in Malta Mediterranean Link Consortium (MMLC) together with Vienna International Airport with 53.24%, and the Bianchi Group, with 10.63%. The MMLC owns 40% of MIA. VIE then owns ...

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