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 SL₂(C), then

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

References

• Jørgensen, Troels (1976), "On discrete groups of Möbius transformations", American Journal of Mathematics, 98 (3): 739–749, doi:10.2307/2373814, ISSN 0002-9327, JSTOR 2373814, MR 0427627