Bianchi group

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For the 3-dimensional Lie groups or Lie algebras, see Bianchi classification.

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

where d is a positive square-free integer. Here, PSL denotes the projective special linear group and is the ring of integers of the imaginary quadratic field .

The groups were first studied by Bianchi (1892) as a natural class of discrete subgroups of , now termed Kleinian groups.

As a subgroup of , a Bianchi group acts as orientation-preserving isometries of 3-dimensional hyperbolic space . The quotient space is a non-compact, hyperbolic 3-fold with finite volume, which is also called Bianchi manifold. An exact formula for the volume, in terms of the Dedekind zeta function of the base field , was computed by Humbert as follows. Let be the discriminant of , and , the discontinuous action on , then

The set of cusps of is in bijection with the class group of . 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

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