zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

20H10Fuchsian groups and their generalizations (group theory)
20F05Generators, relations, and presentations of groups
30F35Fuchsian groups and automorphic functions
11F06Structure of modular groups and generalizations
Full Text: DOI
[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. · Zbl 0459.53031
[4] [BGS]Ballman, W., Gromov, M. & Schroeder, V.,Manifolds of Nonpositive Curvature. Progress in Mathematics, Vol. 61. Birkhäuser, 1985. · Zbl 0591.53001
[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. · Zbl 0628.30027
[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. · Zbl 0722.30034
[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. · Zbl 0738.30032
[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. · Zbl 0001.22703 · 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. · Zbl 0254.22005
[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