Bianchi group

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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 (d). The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of PSL2(), now termed Kleinian groups. As a subgroup of PSL2(), a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space 3. The quotient space Md=PSL2(𝒪d)3 is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi orbifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field (d), was computed by Humbert as follows. Let D be the discriminant of (d), and Γ=SL2(𝒪d), the discontinuous action on , then

vol(Γ)=|D|3/24π2ζ(d)(2).

The set of cusps of Md is in bijection with the class group of (d). It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.[1]

References

  1. Maclachlan & Reid (2003) p. 58
  • Bianchi, Luigi (1892). "Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî". Mathematische Annalen. 40 (3). Springer Berlin / Heidelberg: 332–412. doi:10.1007/BF01443558. ISSN 0025-5831. JFM 24.0188.02. S2CID 120341527.
  • 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. Vol. 129. New York: Marcel Dekker Inc. ISBN 978-0-8247-8192-7. MR 1010229. Zbl 0760.20014.
  • Fine, B. (2001) [1994], "Bianchi group", Encyclopedia of Mathematics, EMS Press
  • Maclachlan, Colin; Reid, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics. Vol. 219. Springer-Verlag. ISBN 0-387-98386-4. Zbl 1025.57001.

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