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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 $$. Then

$max\left\{\parallel {g}^{i}f{g}^{-i}-Id\parallel :\phantom{\rule{1.em}{0ex}}i=0,1,2,···,n\right\}\ge 2-\sqrt{3}·$

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 $SO\left(n+1,1\right)$ which, in the Hilbert-Schmidt norm, has the form $\left\{A\in SO\left(n+1,1\right):$ $\parallel A-E\parallel <2-\sqrt{3}\right\}$. In the non elliptic case, a conjugacy invariant form of the inequality is:

$min\left\{max\left\{\parallel hf{h}^{-1}-Id\parallel ,\phantom{\rule{1.em}{0ex}}\parallel h\left[f,g\right]{h}^{-1}-Id\parallel \right\}:\phantom{\rule{1.em}{0ex}}h\in M\phantom{\rule{1.em}{0ex}}b\left(n\right)\right\}\ge 2-\sqrt{3}·$

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
References:
 [1] [Ah]Ahlfors, L. V.,Möbius transformations in several dimensions. Lecture notes in Mathematics, University of Minnesota, 1981. [2] [Be]Beardon, A.,The Geometry of Discrete Groups. Springer Verlag, 1982. [3] [B.K.]Buser, P. & Karcher, H.,Gromov’s Almost Flat Manifolds. Astérisque 81, 1981. [4] [BGS]Ballman, W., Gromov, M. & Schroeder, V.,Manifolds of Nonpositive Curvature. Progress in Mathematics, Vol. 61. Birkhäuser, 1985. [5] [Ch]Chuchrow, V., On Schottky groups with application to Kleinian groups.Ann of Math., 88 (1968), 47–61. · Zbl 0186.40603 · doi:10.2307/1970555 [6] [G.M.1]Gehring, F. W. &Martin, G. J., Discrete quasiconformal groups I.Proc. London Math. Soc. (3), 55 (1987), 331–358. [7] [G.M.2]— Iteration theory and inequalities for Kleinian groups.Bull. Amer. Math. Soc., 21 (1989), 57–65. · Zbl 0689.30036 · doi:10.1090/S0273-0979-1989-15761-3 [8] [G.M.3]Gehring, F. W. & Martin, G. J., Inequalities for Möbius transformations and discrete groups. To appear. [9] [Jø]Jørgensen, T., On discrete groups of Möbius transformations.Amer. J. Math., 98 (1976), 739–749. · Zbl 0336.30007 · doi:10.2307/2373814 [10] [J.K.]Jørgensen, T. &Klein, P., Algebraic convergence of finitely generated Kleinian groups.Quart. J. Math. Oxford (2), 33 (1982), 325-332. · Zbl 0499.30033 · doi:10.1093/qmath/33.3.325 [11] [J.M.]Jørgensen, T., & Marden, A., Algebraic and geometric convergence of Kleinian groups. To appear. [12] [Ku]Kuratowski, K.,Topology. Academic Press, 1966. [13] [Ma]Marden, A., The geometry of finitely generated Kleinian groups.Ann. of Math., 99 (1974), 383–462. · Zbl 0282.30014 · doi:10.2307/1971059 [14] [Mar]Martin, G. J., Balls in hyperbolic manifolds. To appear inJ. London Math. Soc. [15] [Ne]Newman, M. H. A., A theorem on periodic transformation of spaces.Quart. J. Math. Oxford, 2 (1931), 1–8. · doi:10.1093/qmath/os-2.1.1-a [16] [Ra]Raghunathan, M. S.,Discrete Subgroups of Lie Groups. Ergebnisse, der Mathematik, Vol. 68. Springer-Verlag, 1972. [17] [Sc]Scott, G. P., Finitely generated 3-manifold groups are finitely presented.J. London Math. Soc., 6 (1973), 437–440. · Zbl 0254.57003 · doi:10.1112/jlms/s2-6.3.437 [18] [Se]Selberg, A., On discontinuous groups in higher dimensional symmetric spaces.Contribution to Function Theory. Bombay, 1960, pp. 147–164. [19] [Tu]Tukia, P., On isomorphisms of geometrically finite Möbius groups.Inst. Hautes Études Sci. Publ. Math., 61 (1985), 171–214. · Zbl 0572.30036 · doi:10.1007/BF02698805 [20] [We]Weilenberg, N., Discrete Möbius groups: Fundamental polyhedra and convergence.Amer. J. Math., 99 (1977), 861–867. · Zbl 0373.57024 · doi:10.2307/2373869