In mathematics, a Bianchi group is a group of the form
- PSL_{2}(O_{d})
where d is a positive square-free integer. Here, PSL denotes the projective special linear group and O_{d} 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 PSL_{2}(C), now termed Kleinian groups.
As a subgroup of PSL_{2}(C), a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space H^{3}. The quotient space M_{d} = PSL_{2}(O_{d}) \ 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 Q(√−d), was computed by Humbert. The set of cusps of M_{d} 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]
- ^ 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.
External links[edit]
- Allen Hatcher, Bianchi Orbifolds
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