Bianchi group explained

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

PSL _{2(l{O}} _{d)}

where d is a positive square-free integer. Here, PSL denotes the projective special linear group and

l{O} _{d} is the ring of integers of the imaginary quadratic field Q (\sqrt{ - d}) .

The groups were first studied by 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(l{O}} _{d)} \backslash H ^{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 Q (\sqrt{ - d}) , was computed by Humbert as follows. Let D be the discriminant of Q (\sqrt{ - d}) , and \Gamma = SL _{2(l{O}} _{d)} , the discontinuous action on l{H} , then\operatorname{vol}(\Gamma\backslash H ) = (|D| ^{3/2})/(4 \pi ^{2}) \zeta _{Q(\sqrt{-d} )}( 2 ) .

The set of cusps of

M _{d} is in bijection with the class group of Q (\sqrt{ - d}) . It is well known that every non-cocompact arithmetic Kleinian group is weakly commensurable with a Bianchi group.^[1]

References

• Bianchi . Luigi . Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî . Springer Berlin / Heidelberg . 10.1007/BF01443558 . 1892 . . 0025-5831 . 40 . 3 . 24.0188.02 . 332–412 . 120341527 .
• Book: Juergen . Elstrodt . Fritz . Grunewald . Jens . Mennicke . Groups Acting On Hyperbolic Spaces . . . 1998 . 3-540-62745-6 . 0888.11001 .
• Book: Fine . Benjamin . Algebraic theory of the Bianchi groups . Marcel Dekker Inc. . New York . Monographs and Textbooks in Pure and Applied Mathematics . 978-0-8247-8192-7 . 1010229 . 1989 . 129 . 0760.20014 .
  • Book: Colin . Maclachlan . Alan W. . Reid . The Arithmetic of Hyperbolic 3-Manifolds . . . 219 . 2003 . 0-387-98386-4 . 1025.57001 .

External links

• Allen Hatcher, Bianchi Orbifolds

Notes and References

1. Maclachlan & Reid (2003) p. 58