Jørgensen's inequality

This article is about the mathematical principle. For the principle of Homeric scholarship, see Jørgensen's law.

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

${\displaystyle \left|\operatorname {Tr} (A)^{2}-4\right|+\left|\operatorname {Tr} \left(ABA^{-1}B^{-1}\right)-2\right|\geq 1.\,}$

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

${\displaystyle \left\|A-I\right\|\ \left\|B-I\right\|\geq 1\,}$

where ${\displaystyle \|\cdot \|}$ 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 ${\displaystyle \mathbb {H} ^{3}/G}$. 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

• The Margulis lemma is a qualitative generalisation to more general spaces of negative curvature.

References

1. 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
2. Beardon, Alan F. (1983). The Geometry of Discrete Groups. New York: Springer-Verlag. pp. 104–114. ISBN 9781461211471.
3. Maskit, Bernard (1988). Kleinian Groups. Springer-Verlag. p. 117. ISBN 0387177469.