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Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets


Authors: Petra Bonfert-Taylor and Edward C. Taylor
Journal: Trans. Amer. Math. Soc. 355 (2003), 787-811
MSC (2000): Primary 30C65; Secondary 30F40, 30F45.
DOI: https://doi.org/10.1090/S0002-9947-02-03134-3
Published electronically: October 2, 2002
MathSciNet review: 1932726
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Abstract: We extend the part of Patterson-Sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the Poincaré series to the Hausdorff dimension of the limit set. In doing so we define new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension. We find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of Patterson-Sullivan theory.


References [Enhancements On Off] (What's this?)

  • 1. J. Anderson, P. Bonfert-Taylor, and E. C. Taylor, Convergence groups, Hausdorff dimension, and a theorem of Sullivan and Tukia, preprint, 2002.
  • 2. A. F. Beardon, The geometry of discrete groups, Springer-Verlag, New York, 1983; corrected reprint, 1995. MR 85d:22026; MR 97d:22011
  • 3. C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), 1-39. MR 98k:22043
  • 4. P. Bonfert-Taylor and G. Martin, Discrete quasiconformal groups of compact type, in preparation.
  • 5. P. Bonfert-Taylor and E. C. Taylor, Hausdorff dimension and limit sets of quasiconformal groups, Mich. Math. J. 49 (2001), 243-257. MR 2002g:30018
  • 6. B. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), 245-317. MR 94e:57016
  • 7. M. H. Freedman and R. Skora, Strange actions of groups on spheres, J. Differential Geometry 25 (1987) 75-98. MR 88a:57074
  • 8. F. W. Gehring and G. J. Martin, Discrete quasiconformal groups I, Proc. London Math. Soc. (3) 55 (1987) 331-358. MR 88m:30057
  • 9. F. W. Gehring and G. J. Martin, Discrete quasiconformal groups II, unpublished manuscript.
  • 10. M. Ghamsari, Quasiconformal groups acting on $B\sp 3$that are not quasiconformally conjugate to Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. (2) 20 (1995) 245-250. MR 96h:30034
  • 11. O. Lehto und K. I. Virtannan, Quasikonforme Abbildungen, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 1965. MR 32:5872
  • 12. G. J. Martin, Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), 179-202. MR 89d:30025
  • 13. G. Martin, personal communication.
  • 14. B. Maskit, Kleinian groups, Springer-Verlag, 1988. MR 90a:30132
  • 15. V. Mayer, Cyclic parabolic quasiconformal groups that are not quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. (1) 18 (1993), 147-154. MR 95f:30032
  • 16. P. J. Nicholls, The Ergodic Theory of Discrete Groups, Cambridge University Press, 1989. MR 91i:58104
  • 17. S. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 56:8841
  • 18. D. Stroock, Probability Theory, An Analytic View, Cambridge University Press, 1993. MR 95f:60003
  • 19. D. Sullivan, Hyperbolic geometry and homeomorphisms in geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, 543-555. MR 81m:57012
  • 20. D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES 50 (1979), 171-202. MR 81b:58031
  • 21. D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions in Riemann surfaces and related topics (Proceedings of the 1978 Stony Brook Conference, State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, N.J., 1981, 465-496. MR 83f:58052
  • 22. D. Sullivan, Entropy Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta. Math. 153 (1984), pp. 259-277. MR 86c:58093
  • 23. P. Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. AI Math. 6 (1981), 149-160. MR 83b:30019
  • 24. P. Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. 152 (1984), 127-134. MR 85m:30031
  • 25. P. Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318-346. MR 87m:30043
  • 26. P. Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998), 71-98. MR 2000b:30067
  • 27. P. Tukia and J. Väisälä, Quasiconformal extension from dimension $n$ to $n+1$, Ann. Math. 115 (1982), 331-348. MR 84i:30030
  • 28. J. Väisälä, Lectures on $n$-Dimensional Quasiconformal Mappings, Springer-Verlag, 1971. MR 56:12260

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Additional Information

Petra Bonfert-Taylor
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: pbonfert@wesleyan.edu

Edward C. Taylor
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: ectaylor@wesleyan.edu

DOI: https://doi.org/10.1090/S0002-9947-02-03134-3
Keywords: Kleinian groups, discrete quasiconformal groups, Patterson-Sullivan measure, exponent of convergence, Hausdorff dimension.
Received by editor(s): September 15, 2000
Received by editor(s) in revised form: May 14, 2002
Published electronically: October 2, 2002
Additional Notes: The first author was supported in part by NSF grant 0070335
The second author was supported in part by an NSF Postdoctoral Fellowship
Article copyright: © Copyright 2002 American Mathematical Society

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