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On discrete Möbius groups in all dimensions: A generalization of Jørgensen’s inequality. (English) Zbl 0698.20037

The author presents a generalization to all dimensions of T. Jørgensen’s inequality [Am. J. Math. 98, 739-749 (1976; Zbl 0336.30007)] in the following form: Theorem. Let f and g be Möbius transformations of S n that generate a discrete non elementary group <f,g>. Then

max{g i fg -i -Id:i=0,1,2,···,n}2-3·

Moreover, if f is non elliptic then it suffices to consider only those terms with i=0 or i=1. Here the author’s main concept is the calculation of the Zassenhaus neighbourhood shape in SO(n+1,1) which, in the Hilbert-Schmidt norm, has the form {ASO(n+1,1): A-E<2-3}. In the non elliptic case, a conjugacy invariant form of the inequality is:

min{max{hfh -1 -Id,h[f,g]h -1 -Id}:hMb(n)}2-3·

Reviewer: B.N.Apanasov

20H10Fuchsian groups and their generalizations (group theory)
20F05Generators, relations, and presentations of groups
30F35Fuchsian groups and automorphic functions
11F06Structure of modular groups and generalizations
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