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Elliptic elements in Möbius groups

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Abstract

Let G be a nonelementary subgroup of Isom(H n). In this paper we prove that if G contains elliptic elements and dim(M G) is even, then G is discrete if and only if WY (G) is discrete and every nonelementary subgroup of G generated by two elliptic elements is discrete. We also describe an example to show the assumption on dim(M G) is necessary.

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References

  1. S. Chen and L. Greenberg, Hyperbolic spaces, in Contributions to Analysis (A collection of papers dedicated to L. Bers), Academic Press, 1974.

  2. L. Greenberg, Discrete subgroups of the Lorentz group, Mathematica Scandinavica 10 (1962), 85–107.

    Article  MathSciNet  Google Scholar 

  3. J. Gilman, Inequalities in discrete subgroups of PSL(2,R), Canadian Journal of Mathematics 40 (1988), 115–130.

    Article  MathSciNet  Google Scholar 

  4. N. A. Isokenko, Systems of generators of subgroups of PSL(2,C), Siberian Mathematical Journal 31 (1990), 162–165.

    Article  MathSciNet  Google Scholar 

  5. T. Jørgensen, On discrete groups of Möbius transformations, American Journal of Mathematics 98 (1976), 739–749.

    Article  MathSciNet  Google Scholar 

  6. G. Martin, On discrete isometry groups of negative curvature, Pacific Journal of Mathematics 160 (1993), 109–127.

    Article  MathSciNet  Google Scholar 

  7. G. Rosenberger, Minimal generating systems of subgroups of SL(2,C), Proceedings of the Edinburgh Mathematical Society 31 (1988), 261–265.

    Article  MathSciNet  Google Scholar 

  8. P. Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand Journal of Mathematics 23 (1994), 157–187.

    MathSciNet  MATH  Google Scholar 

  9. P. Tukia and X. Wang, Discreteness of subgroups of SL(2,C) containing elliptic elements, Mathematica Scandinavica 91 (2002), 214–220.

    Article  MathSciNet  Google Scholar 

  10. X. Wang, L. Li and W. Cao, Discreteness criteria for Möbius groups acting on R n, Israel Journal of Mathematics 150 (2005), 357–368.

    Article  MathSciNet  Google Scholar 

  11. P. Waterman, Purely elliptic Möbius groups, Holomorphic functions and moduli, II (1988), 173–178.

    Article  Google Scholar 

  12. S. Yang, On the discreteness criterion in SL(2,C), Mathematische Zeitschrift 255 (2007), 227–230.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai University of Finance and Economics, 777 Gouding Road, Shanghai, 200433, P. R. China

    Shihai Yang

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  1. Shihai Yang
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Correspondence to Shihai Yang.

Additional information

Supported in part by NSFC 10671059 and by Leading academic Discipline program, 211 project for Shanghai University of Finance and Economics (the 3rd phase).

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Yang, S. Elliptic elements in Möbius groups. Isr. J. Math. 172, 309–315 (2009). https://doi.org/10.1007/s11856-009-0074-1

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  • DOI: https://doi.org/10.1007/s11856-009-0074-1

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