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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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3 news items

 
Times of Malta
Sun, 19 Oct 2014 23:00:00 -0700

Ltd) and the Timmi Ritscher from Damietta to Genoa (Bianchi Group) tomorrow. The Maersk Edison from Fos Sur Mer to Port Said (Thomas Smith Shipping Ltd) on Wednesday. The Agamemnon from Port Said to Genoa (Thomas Smith Shipping Ltd) on ...

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 ...

La Provincia di Como

La Provincia di Como
Sun, 05 Oct 2014 03:52:30 -0700

Ma saranno le racchette dei tennisti paraplegici a chiudere la stagione con il Bianchi Group Trophy, appuntamento internazionale in carrozzina che si terrà tra i campi di Casnate e Villaguardia dal 9 al 12 ottobre con campioni quali Diego Amadori e ...
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