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Jørgensen's inequality
Inequality involving the traces of elements of a Kleinian group From Wikipedia, the free encyclopedia
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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
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
where 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 . 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]
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See also
- The Margulis lemma is a qualitative generalisation to more general spaces of negative curvature.
References
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