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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 {\it 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 $S\sp n$ that generate a discrete non elementary group $<f,g>$. Then $$\max \{\Vert g\sp ifg\sp{-i}-Id\Vert:\quad i=0,1,2,...,n\}\ge 2-\sqrt{3}.$$ 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 $\{A\in SO(n+1,1):$ $\Vert A-E\Vert <2-\sqrt{3}\}$. In the non elliptic case, a conjugacy invariant form of the inequality is: $$\min \{\max \{\Vert hfh\sp{-1}-Id\Vert,\quad \Vert h[f,g]h\sp{-1}-Id\Vert \}:\quad h\in M\quad b(n)\}\ge 2-\sqrt{3}.$$
Reviewer: B.N.Apanasov

##### MSC:
 20H10 Fuchsian groups and their generalizations (group theory) 20F05 Generators, relations, and presentations of groups 30F35 Fuchsian groups and automorphic functions 11F06 Structure of modular groups and generalizations
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##### References:
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