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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

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[edit]

  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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44 news items

 
Times of Malta
Sun, 24 May 2015 22:26:15 -0700

The MSC Faustina from Port Said to Valencia (John Ripard & Son Ltd) and the Jakarta Express from Damietta to Genoa (Bianchi Group) on Wednesday. The Laura Ann from Tunis to Tunis, the Eclipse from Annaba to Algiers, the K-River from Algiers to Tunis ...
 
MaltaToday
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 ...

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 ...

Affaritaliani.it

Affaritaliani.it
Tue, 13 Jan 2015 02:52:30 -0800

Casta Diva Group, holding capitanata da Andrea De Micheli e Luca Oddo, rispettivamente amministratore delegato e presidente – ha scelto l'agenzia Alessia Bianchi Group come partner al quale affidare la gestione del rapporto con i media. “Sono molto ...
 
MaltaToday
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 ...
 
MaltaToday
Wed, 25 Jun 2014 06:17:02 -0700

Malta International Airport has announced the resignation of Michael Bianchi, of the Bianchi Group, citing personal reasons. As a director of the MIA, Bianchi represented the Bianchi Group, which owns a 10.63% stake in the Malta Mediterranean Link ...
 
Bike Biz
Mon, 17 Nov 2014 02:37:21 -0800

First stocks of Italy-based Velomann (part of the Bianchi Group) has arrived with new distributor Greyville Enterprises. The Velomann range, comprising lights, cycle computers and home trainer, is pitched at the mid to higher end of the market. Using ...

SpeedLux

SpeedLux
Fri, 21 Nov 2014 08:41:15 -0800

Keeping in mind Bianchi's group Marussia has been disbanded because of budgetary inconveniences, Caterham, which has monetary inconveniences of its own, has figured out how to raise enough supports to make it out to Abu Dhabi for this last round.
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