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 \[ \max \{\| g^ ifg^{-i}-Id\|:\quad i=0,1,2,...,n\}\geq 2-\sqrt{3}. \] 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 \(\{A\in SO(n+1,1):\) \(\| A-E\| <2-\sqrt{3}\}\). In the non elliptic case, a conjugacy invariant form of the inequality is: \[ \min \{\max \{\| hfh^{-1}-Id\|,\quad \| h[f,g]h^{-1}-Id\| \}:\quad h\in M\quad b(n)\}\geq 2-\sqrt{3}. \]
Reviewer: B.N.Apanasov


20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11F06 Structure of modular groups and generalizations; arithmetic groups


Zbl 0336.30007
Full Text: DOI


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