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). The inequality states that if A and B generate a non-elementary discrete subgroup of the SL2(C), then
| Tr ( A ) 2 − 4 | + | Tr ( A B A − 1 B − 1 ) − 2 | ≥ 1. {\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
‖ A − I ‖ ‖ B − I ‖ ≥ 1 {\displaystyle \left\|A-I\right\|\ \left\|B-I\right\|\geq 1\,}
where ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denotes the usual norm on SL2(C). 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 H 3 / G {\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.
See also The Margulis lemma is a qualitative generalisation to more general spaces of negative curvature.
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