Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets
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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
- J. Anderson, P. Bonfert-Taylor, and E. C. Taylor, Convergence groups, Hausdorff dimension, and a theorem of Sullivan and Tukia, preprint, 2002.
- Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
- Christopher J. Bishop and Peter W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), no. 1, 1–39. MR 1484767, DOI 10.1007/BF02392718
- P. Bonfert-Taylor and G. Martin, Discrete quasiconformal groups of compact type, in preparation.
- Petra Bonfert-Taylor and Edward C. Taylor, Hausdorff dimension and limit sets of quasiconformal groups, Michigan Math. J. 49 (2001), no. 2, 243–257. MR 1852301, DOI 10.1307/mmj/1008719771
- B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245–317. MR 1218098, DOI 10.1006/jfan.1993.1052
- Michael H. Freedman and Richard Skora, Strange actions of groups on spheres, J. Differential Geom. 25 (1987), no. 1, 75–98. MR 873456
- F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224, DOI 10.1093/plms/s3-55_{2}.331
- F. W. Gehring and G. J. Martin, Discrete quasiconformal groups II, unpublished manuscript.
- Manouchehr Ghamsari, Quasiconformal groups acting on $B^3$ that are not quasiconformally conjugate to Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 245–250. MR 1346809
- O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin-New York, 1965 (German). MR 0188434, DOI 10.1007/978-3-662-42594-7
- Gaven J. Martin, Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 2, 179–202. MR 853955, DOI 10.5186/aasfm.1986.1113
- G. Martin, personal communication.
- Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
- Volker Mayer, Cyclic parabolic quasiconformal groups that are not quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 1, 147–154. MR 1207901
- Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
- S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
- Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569
- Dennis Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 543–555. MR 537749
- Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586, DOI 10.1007/BF02684773
- Dennis 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 (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496. MR 624833
- Dennis Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3-4, 259–277. MR 766265, DOI 10.1007/BF02392379
- Pekka Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 1, 149–160. MR 639972, DOI 10.5186/aasfm.1981.0625
- Pekka Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. 152 (1984), no. 1-2, 127–140. MR 736215, DOI 10.1007/BF02392194
- Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318–346. MR 861709, DOI 10.1007/BF02796595
- Pekka Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998), 71–98. MR 1637829, DOI 10.1515/crll.1998.081
- P. Tukia and J. Väisälä, Quasiconformal extension from dimension $n$ to $n+1$, Ann. of Math. (2) 115 (1982), no. 2, 331–348. MR 647809, DOI 10.2307/1971394
- Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
Additional Information
- Petra Bonfert-Taylor
- Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
- MR Author ID: 617474
- Email: pbonfert@wesleyan.edu
- Edward C. Taylor
- Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
- Email: ectaylor@wesleyan.edu
- 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 - © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1932726