×

zbMATH — the first resource for mathematics

Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets. (English) Zbl 1013.30024
The authors extend some aspects of the Patterson-Sullivan theory, that is, the construction of a class of finite positive measures on the limit set of a Kleinian group and the relation between the exponent of convergence of the Poincaré series of the action and the Hausdorff dimension of the limit set. The extension concerns the setting of quasiconformal Fuchsian groups, that is, discrete groups of uniformly \(K\)-quasi-conformal mappings preserving the closed unit ball \(B^n\). In doing so, the authors define some new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension.

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Anderson, P. Bonfert-Taylor, and E. C. Taylor, Convergence groups, Hausdorff dimension, and a theorem of Sullivan and Tukia, preprint, 2002. · Zbl 1067.30041
[2] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. · Zbl 0528.30001
[3] Christopher J. Bishop and Peter W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), no. 1, 1 – 39. · Zbl 0921.30032 · doi:10.1007/BF02392718 · doi.org
[4] P. Bonfert-Taylor and G. Martin, Discrete quasiconformal groups of compact type, in preparation. · Zbl 1281.30033
[5] Petra Bonfert-Taylor and Edward C. Taylor, Hausdorff dimension and limit sets of quasiconformal groups, Michigan Math. J. 49 (2001), no. 2, 243 – 257. · Zbl 0999.30029 · doi:10.1307/mmj/1008719771 · doi.org
[6] B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245 – 317. · Zbl 0789.57007 · doi:10.1006/jfan.1993.1052 · doi.org
[7] Michael H. Freedman and Richard Skora, Strange actions of groups on spheres, J. Differential Geom. 25 (1987), no. 1, 75 – 98. · Zbl 0588.57024
[8] F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331 – 358. · Zbl 0628.30027 · doi:10.1093/plms/s3-55_2.331 · doi.org
[9] F. W. Gehring and G. J. Martin, Discrete quasiconformal groups II, unpublished manuscript. · Zbl 0628.30027
[10] Manouchehr Ghamsari, Quasiconformal groups acting on \?³ that are not quasiconformally conjugate to Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 245 – 250. · Zbl 0851.30008
[11] O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band, Springer-Verlag, Berlin-New York, 1965 (German). · Zbl 0138.30301
[12] 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. · Zbl 0635.30021 · doi:10.5186/aasfm.1986.1113 · doi.org
[13] G. Martin, personal communication.
[14] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. · Zbl 0627.30039
[15] 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. · Zbl 0781.30017
[16] Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. · Zbl 0674.58001
[17] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241 – 273. · Zbl 0336.30005 · doi:10.1007/BF02392046 · doi.org
[18] Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. · Zbl 0925.60004
[19] Dennis Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 543 – 555.
[20] Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171 – 202. · Zbl 0439.30034
[21] 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.
[22] 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. · Zbl 0566.58022 · doi:10.1007/BF02392379 · doi.org
[23] 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. · Zbl 0443.30026 · doi:10.5186/aasfm.1981.0625 · doi.org
[24] Pekka Tukia, The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. 152 (1984), no. 1-2, 127 – 140. · Zbl 0539.30034 · doi:10.1007/BF02392194 · doi.org
[25] Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318 – 346. · Zbl 0603.30026 · doi:10.1007/BF02796595 · doi.org
[26] Pekka Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998), 71 – 98. · Zbl 0909.30034 · doi:10.1515/crll.1998.081 · doi.org
[27] P. Tukia and J. Väisälä, Quasiconformal extension from dimension \? to \?+1, Ann. of Math. (2) 115 (1982), no. 2, 331 – 348. · Zbl 0484.30017 · doi:10.2307/1971394 · doi.org
[28] Jussi Väisälä, Lectures on \?-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. · Zbl 0221.30031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.