In the mathematical theory of Kleinian groups, Jørgensen's inequality is an inequality involving the traces of elements of a Kleinian group, proved by Troels Jørgensen (1976).[1]
The inequality states that if A and B generate a non-elementary discrete subgroup of the SL2(C), then
The inequality gives a quantitative estimate of the discreteness of the group: many of the standard corollaries bound elements of the group away from the identity. For instance, if A is parabolic, then
where denotes the usual norm on SL2(C).[2]
Another consequence in the parabolic case is the existence of cusp neighborhoods in hyperbolic 3-manifolds: if G is a Kleinian group and j is a parabolic element of G with fixed point w, then there is a horoball based at w which projects to a cusp neighborhood in the quotient space . Jørgensen's inequality is used to prove that every element of G which does not have a fixed point at w moves the horoball entirely off itself and so does not affect the local geometry of the quotient at w; intuitively, the geometry is entirely determined by the parabolic element.[3]
See also
edit- The Margulis lemma is a qualitative generalisation to more general spaces of negative curvature.
References
edit- ^ Jørgensen, Troels (1976), "On discrete groups of Möbius transformations", American Journal of Mathematics, 98 (3): 739–749, doi:10.2307/2373814, ISSN 0002-9327, JSTOR 2373814, MR 0427627
- ^ Beardon, Alan F. (1983). The Geometry of Discrete Groups. New York: Springer-Verlag. pp. 104–114. ISBN 9781461211471.
- ^ Maskit, Bernard (1988). Kleinian Groups. Springer-Verlag. p. 117. ISBN 0387177469.