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Triangle subgroups of Kleinian groups

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Commentarii Mathematici Helvetici

Abstract

We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF 2=∏/Γ ∏ is always embedded in the hyperbolic 3-orbifoldM 3=ℍ3/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup.

We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation of co-volumes of Kleinian groups containing these triangle subgroups.

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References

  1. C. C. Adams,Limit volumes of hyperbolic 3-orbifolds, J. Diff. Geom.34 (1991) 115–141.

    MATH  Google Scholar 

  2. A. Beardon,The geometry of discrete groups, Springer-Verlag, 1983.

  3. M. D. E. Conder andG. J. Martin,Cusps, triangle groups and hyperbolic 3-folds, J. Australian Math. Soc.55 (1993) 149–182.

    MathSciNet  MATH  Google Scholar 

  4. F. W. Gehring andG. J. Martin,Inequalities for Möbius transformations and discrete groups, J. reine angew. Math.418 (1991) 31–76.

    MathSciNet  MATH  Google Scholar 

  5. F. W. Gehring andG. J. Martin,Commutators, collars and the geometry of Möbius groups, J. d'Analyse Math63 (1994) 175–219.

    Article  MathSciNet  MATH  Google Scholar 

  6. F. W. Gehring andG. J. Martin,Kleinian groups with an elliptic generator, to appear.

  7. F. W. Gehring andG. J. Martin,n-dimensional collaring theorems and the volume of symmetric hyperbolic n-manifolds, to appear.

  8. F. W. Gehring andG. J. Martin, 6-Torsion and hyperbolic volume, Proc. Amer. Math. Soc.117 (1993) 727–735.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. W. Gehring andG. J. Martin,On the hyperbolic 3-fold of minimal co-volume (with F. W. Gehring), Mathematical Research Letters1 (1994) 107–115.

    MathSciNet  MATH  Google Scholar 

  10. F. W. Gehring, C. Maclachlan, G. J. Martin andA. Reid,Arithmeticity, Discreteness and Volume, Trans. Amer. Math. Soc.

  11. T. Jørgensen,On discrete groups of Möbius transformations, Amer. J. Math.98 (1976) 739–749.

    Article  MathSciNet  Google Scholar 

  12. B. Maskit,Kleinian groups, Springer-Verlag, 1987.

  13. T. Marshall,Hyperbolic Geometry and Reflection Groups, PhD Thesis, The University of Auckland, 1994.

  14. W. P. Thurston,The Geometry and Topology of 3-manifolds, Princeton University Lecture Notes, 1976.

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Authors and Affiliations

  1. Centre for Mathematics & Applications, Australian National University, Canberra, ACT, Australia

    Gaven J. Martin

  2. Department of Mathematics, The University of Auckland, Auckland, New Zealand

    Gaven J. Martin

Authors
  1. Gaven J. Martin
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Additional information

for J. A. Kalman on the occasion of his 65th birthday

Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science and Technology and the U.K. Scientific and Engineering Research Council.

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Martin, G.J. Triangle subgroups of Kleinian groups. Commentarii Mathematici Helvetici 71, 339–361 (1996). https://doi.org/10.1007/BF02566424

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  • DOI: https://doi.org/10.1007/BF02566424

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