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In mathematics, a Bianchi group is a group of the form

PSL2(Od)

where d is a positive square-free integer. Here, PSL denotes the projective special linear group and Od is the ring of integers of the imaginary quadratic field Q(√−d).

The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of PSL2(C), now termed Kleinian groups.

As a subgroup of PSL2(C), a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H3. The quotient space Md = PSL2(Od) \ H3 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 Q(√−d), was computed by Humbert. The set of cusps of Md is in bijection with the class group of Q(√−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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2 news items

 
Times of Malta
Sun, 30 Mar 2014 22:48:45 -0700

The Jaguar from Sfax to Misurata, the Rosa from Asuncion to Haifa (both Thomas Smith Shipping Ltd) and the Northern Dedocation from Valencia to Livorno (Bianchi Group) today. The Devon Strait from Gemlik, the CMA CGM Christophe Colomb from Le ...
 
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Tue, 22 Apr 2014 01:53:58 -0700

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