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Quasiconformal groups with small dilatation I


Authors: Petra Bonfert-Taylor and Gaven Martin
Journal: Proc. Amer. Math. Soc. 129 (2001), 2019-2029
MSC (1991): Primary 30F40, 57S30, 30C65, 20H10
DOI: https://doi.org/10.1090/S0002-9939-00-05765-8
Published electronically: November 30, 2000
MathSciNet review: 1825913
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Abstract:

We study Fuchsian quasiconformal groups with small dilatation. For this class of groups we establish a Jørgensen-type inequality in all dimensions. We show discreteness persists to the limit under algebraic convergence and that such groups are discrete if and only if every two generator subgroup is discrete.


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  • [Bea] A.F. Beardon, The geometry of discrete groups Springer-Verlag, New York, 1983. MR 97d:22011
  • [BT] P. Bonfert-Taylor, Jørgensen's inequality for discrete convergence groups, Ann. Acad. Sci. Fenn. Math. 25 (2000) 131-150.
  • [CaWa] C. Cao and P. Waterman, Conjugacy invariants of Möbius groups, Quasiconformal mappings and analysis, Springer-Verlag, 1997. MR 98k:30058
  • [FrHe] S. Friedland and S. Hersonsky, Jørgensen's inequality for discrete groups in normed algebras, Duke Math. J. 69 (1993) 593-614. MR 94c:46100
  • [FrSk] M.H. Freedman and R. Skora, Strange actions of groups on spheres, J. Differential Geom. 25 (1987) 75-98. MR 88a:57074
  • [Geh] F.W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962) 353-393. MR 25:3166
  • [GeMa] F.W. Gehring and G.J. Martin, Discrete quasiconformal groups I, Proc. London Math. Soc. (3) 55 (1987) 331-358. MR 88m:30057
  • [Her] S. Hersonsky, A generalization of Shimitzu-Leutbecher and Jørgensen's inequalities to Möbius transformation in $\mathbb{R} ^n$. Proc. Amer. Math. Soc. 121 (1994) 209-215. MR 94m:30085
  • [Jør] T. Jørgenson, On discrete groups of Möbius transformations, Amer. J. Math. 96 (1976) 739-749.
  • [Mar1] G.J. Martin, Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. AI Math. 11 (1986) 179-202. MR 89d:30025
  • [Mar2] G.J. Martin, Discrete Möbius groups in all dimensions: A generalization of Jørgensen's inequality, Acta Math. 163 (1989) 253-289. MR 91g:30049
  • [Mar3] G.J. Martin, Quasiconformal and affine groups. J. Diff. Geom. 29 (1989) 427-448. MR 90a:30064
  • [Mar4] G.J. Martin, Algebraic convergence of discrete isometry groups of negative curvature, Pacific J. Math. 160 (1992) 109-127.
  • [McK] M.J.M. McKemie, Quasiconformal groups with small dilatation, Ann. Acad. Sci. Fenn. Ser. AI Math. 12 (1987) 95-118. MR 88j:30045
  • [Sul] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: proceedings of the 1978 Stony Brook Conference, Annals of Mathematics Studies 97, Princeton University Press, 1981, pp. 465-496. MR 83f:58052
  • [Tuk1] P. Tukia, On two dimensional quasiconformal groups, Ann. Acad. Sci. Fenn. 5 (1980) 73-78. MR 82c:30031
  • [Tuk2] P. Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. AI Math. 10 (1985) 561-562.
  • [Tuk3] P. Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157-187. MR 96c:30042
  • [Vai] J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Springer-Verlag, Berlin, 1971. MR 56:12260
  • [Wat] P. Waterman, Möbius transformations in several dimensions, Adv. Math. 101 (1993) 87-113. MR 95h:30056

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Additional Information

Petra Bonfert-Taylor
Affiliation: Department of Mathematics, The University of Michigan, Ann Arbor, Michigan 48109-1109
Address at time of publication: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: bonfert@math.lsa.umich.edu, pbonfert@wesleyan.edu

Gaven Martin
Affiliation: Department of Mathematics, University of Auckland, New Zealand
Email: martin@math.auckland.ac.nz

DOI: https://doi.org/10.1090/S0002-9939-00-05765-8
Received by editor(s): April 16, 1999
Received by editor(s) in revised form: November 9, 1999
Published electronically: November 30, 2000
Additional Notes: The first author acknowledges research support in part by a University of Michigan Rackham fellowship.
The second author acknowledges research support in part by a grant from the NZ Marsden Fund.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

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