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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 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[edit]

  1. ^ Maclachlan & Reid (2003) p.58

External links[edit]



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Times of Malta
Sun, 23 Mar 2014 23:56:15 -0700

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