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 is the ring of integers of the imaginary quadratic field .
The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of , now termed Kleinian groups.
As a subgroup of , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space . The quotient space 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 , was computed by Humbertas follows. Let be the discriminant of , and , the discontinuous action on , then
The set of cusps of is in bijection with the class group of . It is well known that any non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.
- ^ Maclachlan & Reid (2003) p.58
- Bianchi, Luigi (1892). "Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî". Mathematische Annalen (Springer Berlin / Heidelberg) 40: 332–412. doi:10.1007/BF01443558. ISSN 0025-5831. JFM 24.0188.02.
- Elstrodt, Juergen; Grunewald, Fritz; Mennicke, Jens (1998). Groups Acting On Hyperbolic Spaces. Springer Monographs in Mathematics. Springer Verlag. ISBN 3-540-62745-6. Zbl 0888.11001.
- Fine, Benjamin (1989). Algebraic theory of the Bianchi groups. Monographs and Textbooks in Pure and Applied Mathematics 129. New York: Marcel Dekker Inc. ISBN 978-0-8247-8192-7. MR 1010229. Zbl 0760.20014.
- Fine, B. (2001), "Bianchi group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.
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.
Tue, 07 Oct 2014 17:19:47 -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 ...
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 ...
Times of Malta
Sun, 16 Nov 2014 23:18:45 -0800
Ltd) the Lahore Express from Daimetta to Genoa (Bianchi Group) and the Antwerp from Bejaia to Bejaia (Thomas Smith Shipping Ltd) tomorrow. The MV Hoegh Trader from Kobe to Southampton (Sullivan Maritime) and the Maersk Emden from Fos-Sur-Mer to ...
Mon, 17 Nov 2014 02:30:00 -0800
First stocks of Italy-based Velomann (part of the Bianchi Group) has arrived with new distributor Greyville Enterprises. The Velomann range, comprising lights, cycle computers and home trainer, is pitched at the mid to higher end of the market. Using ...
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 ...
Tue, 14 Oct 2014 07:26:32 -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 ...
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 ...
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