Jørgensen's inequality explained

In the mathematical theory of Kleinian groups, Jørgensen's inequality is an inequality involving the traces of elements of a Kleinian group, proved by .

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 boundelements of the group away from the identity. For instance, if A is parabolic,then

where denotes the usual norm on SL₂(C).^[1]

Another consequence in the parabolic case is the existence of cusp neighborhoods in hyperbolic 3-manifolds: if G is aKleinian group and j is a parabolic element of G with fixed point w, then there is a horoball based at wwhich projects to a cusp neighborhood in the quotient space

. Jørgensen's inequalityis used to prove that every element of G which does not have a fixed point at w moves the horoball entirelyoff itself and so does not affect the local geometry of the quotient at w; intuitively, the geometry is entirely determinedby the parabolic element.^[2]

See also

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

Notes and References

1. Book: Beardon . Alan F. . The Geometry of Discrete Groups . 1983 . Springer-Verlag . New York . 9781461211471 . 104–114.
2. Book: Maskit . Bernard . Kleinian Groups . 1988 . Springer-Verlag . 0387177469 . 117.