Uniform perfectness of the limit sets of Kleinian groups

Sugawa, Toshiyuki

doi:10.1090/S0002-9947-01-02775-1

Uniform perfectness of the limit sets of Kleinian groups
HTML articles powered by AMS MathViewer

by Toshiyuki Sugawa

Trans. Amer. Math. Soc. 353 (2001), 3603-3615

DOI: https://doi.org/10.1090/S0002-9947-01-02775-1

Published electronically: May 4, 2001

PDF | Request permission

Abstract:

In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), no. 3, 475–483. MR 518232, DOI 10.1112/jlms/s2-18.3.475
Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. MR 297992, DOI 10.2307/1970638
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
Richard D. Canary, The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), no. 2, 349–359. MR 1136380, DOI 10.1215/S0012-7094-91-06417-3
P. Järvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. (2) 54 (1996), no. 3, 515–529. MR 1413895, DOI 10.1112/jlms/54.3.515
Bernard Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386. MR 802500, DOI 10.5186/aasfm.1985.1042
Katsuhiko Matsuzaki, Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces, Geometric complex analysis (Hayama, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 443–450. MR 1453626
C. McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990), no. 2, 425–454. MR 1031909, DOI 10.1007/BF01234427
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Douglas Niebur and Mark Sheingorn, Characterization of Fuchsian groups whose integrable forms are bounded, Ann. of Math. (2) 106 (1977), no. 2, 239–258. MR 466533, DOI 10.2307/1971094
Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32 (1979), no. 2, 192–199. MR 534933, DOI 10.1007/BF01238490
Ch. Pommerenke, On uniformly perfect sets and Fuchsian groups, Analysis 4 (1984), no. 3-4, 299–321. MR 780609, DOI 10.1524/anly.1984.4.34.299
P. Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces, J. Differential Geom. 52 (1999), 407–452.
R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS’s, Proc. Amer. Math. Soc. 128 (2000), 2569–2575.
T. Sugawa, On the geometry of hyperbolic $2$-orbifolds, preprint
Toshiyuki Sugawa, Various domain constants related to uniform perfectness, Complex Variables Theory Appl. 36 (1998), no. 4, 311–345. MR 1670075, DOI 10.1080/17476939808815116
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
Pekka Tukia, On limit sets of geometrically finite Kleinian groups, Math. Scand. 57 (1985), no. 1, 29–43. MR 815428, DOI 10.7146/math.scand.a-12104