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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. An exact formula for the volume, in terms of the Dedekind zeta function of the base field \mathbb{Q}(\sqrt{-d}), was computed by Humbertas 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]



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50 news items

Daily Mail

Daily Mail
Fri, 10 Oct 2014 03:38:18 -0700

Jules Bianchi's father has revealed there is no change in his son's condition as he continues to fight for his life in hospital. Bianchi's brother Tom and his sister Melanie arrived at the Mie University Hospital late on Thursday where the Frenchman is ...

Daily Mail

Daily Mail
Tue, 07 Oct 2014 09:52:30 -0700

Formula One driver Jules Bianchi remains in a "critical but stable" condition after suffering a traumatic and devastating brain injury in a crash during Sunday's Japanese Grand Prix. It is still too early to say if, and to what extent, Bianchi will ...
 
Times of Malta
Sun, 23 Nov 2014 23:18:45 -0800

Ltd), the Maersk Brooklyn from Antwerp to Haifa (Thomas Smith Shipping Ltd) and the Jakarta Express from Damietta to Genoa (Bianchi Group) tomorrow. The Maersk Sofia from Port Tanger to Port Said (Thomas Smith Shipping Ltd) on Wednesday.

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.

Zee News

Zee News
Tue, 14 Oct 2014 07:26:15 -0700

The father of stricken Formula One driver Jules Bianchi has revealed the family`s anguish as they continue to hold a bedside vigil in the hope he will "win the biggest qualifying lap of his life." Marussia team driver Bianchi remains in a "stable but ...
 
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 ...
 
Times LIVE
Tue, 14 Oct 2014 05:39:01 -0700

Family members of injured Marussia driver Jules Bianchi of France, (L-R) brother Tom, mother Christine, sister Melanie and father Philippe, arrive at the front yard of the Mie General Medical Centre in Yokkaichi on October 10, 2014. Horrifying footage ...

Daily Mail

Daily Mail
Tue, 28 Oct 2014 03:22:38 -0700

Diana Bianchi has now married food writer Aaron Arizpe from New York, and changed her last name. 'She was just a kid when [her affair with Cook] happened,' a family friend told the New York Post.
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