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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]

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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 ...


Tue, 13 Jan 2015 02:52:30 -0800

Il 2015 porta altre novità in Alessia Bianchi Group: una corporate image completamente rinnovata (fatto salvo per il famoso logo “gallopardo”), un nuovo sito e l'apertura della pagina Facebook sulla quale verranno pubblicati i lavori più interessanti ...
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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 ...
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 ...


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Tue, 04 Sep 2012 07:53:08 -0700

“Essentially, what it's done is change the way some of the 2006 penalties are calculated so that small employers aren't getting overly taxed,” said Alden Bianchi, group leader of the employee benefits and executive compensation practice at Boston-based ...

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