Jørgensen's inequality
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
- [math]\displaystyle{ \left|\operatorname{Tr}(A)^2 -4\right| + \left|\operatorname{Tr}\left(ABA^{-1}B^{-1}\right)-2\right|\ge 1. \, }[/math]
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
- [math]\displaystyle{ \left\|A - I\right\|\ \left\|B - I\right\|\ge 1 \, }[/math]
where [math]\displaystyle{ \|\cdot\| }[/math] 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 [math]\displaystyle{ \mathbb{H}^3/G }[/math]. 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
- ↑ Jørgensen, Troels (1976), "On discrete groups of Möbius transformations", American Journal of Mathematics 98 (3): 739–749, doi:10.2307/2373814, ISSN 0002-9327
- ↑ 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.
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