Bianchi group
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
- ↑ 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
- Elstrodt, Juergen; Grunewald, Fritz; Mennicke, Jens (1998). Groups Acting On Hyperbolic Spaces. Springer Monographs in Mathematics. Springer. ISBN 3-540-62745-6.
- 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
- 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
- Allen Hatcher, Bianchi Orbifolds