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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]



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

Yahoo News

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Tue, 07 Oct 2014 08:18:48 -0700

French Formula One driver Jules Bianchi remains in a critical but stable condition in hospital in Japan after suffering a traumatic brain injury, a joint statement by his family and the hospital said on Tuesday.

Daily Mail

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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 ...

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