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).

The inequality states that if A and B generate a non-elementary discrete subgroup of the SL₂(C), then

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

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

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

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