Jørgensen's inequality

In hyperbolic geometry, a branch of mathematics, the Jørgensen 's inequality is a necessary condition for the discreteness of groups of isometries of the 3-dimensional hyperbolic space.

Inequality

It is a non- elementary Kleinian group generated by two matrices, then the inequality holds

Where the trace of a matrix and the commutator of two matrices respectively.

Said clear the condition that the two elements, which produce a non- elementary discrete group, can not be too close to the identity.

Applications

• The Jørgensen 's inequality is used in many convergence proofs in the theory of Kleinian groups.
• Jørgensen's original application was the proof of the following convergence theorem: Let be a non- elementary Kleinian group and a sequence of isomorphisms, which converges to a homomorphism, then is a Kleinian group and is an isomorphism.
• If is parabolic, we obtain the classical result on the existence of invariant precisely Horosphären.
• There are numerous generalizations of Jørgensen 's inequality for discrete groups of isometries of other metric spaces.