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Uniform perfectness of the limit sets of Kleinian groups
Author:
Toshiyuki Sugawa
Journal:
Trans. Amer. Math. Soc. 353 (2001), 3603-3615
MSC (2000):
Primary 30F40; Secondary 30F45
Posted:
May 4, 2001
MathSciNet review:
1837250
Full-text PDF Free Access
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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.
- 1.
Lars
V. Ahlfors, Lectures on quasiconformal mappings, Manuscript
prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand
Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New
York-London, 1966. MR 0200442
(34 #336)
- 2.
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 (80a:30020), http://dx.doi.org/10.1112/jlms/s2-18.3.475
- 3.
Lipman
Bers, On boundaries of Teichmüller spaces and on Kleinian
groups. I, Ann. of Math. (2) 91 (1970),
570–600. MR 0297992
(45 #7044)
- 4.
Christopher
J. Bishop and Peter
W. Jones, Hausdorff dimension and Kleinian groups, Acta Math.
179 (1997), no. 1, 1–39. MR 1484767
(98k:22043), http://dx.doi.org/10.1007/BF02392718
- 5.
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
(92k:57020), http://dx.doi.org/10.1215/S0012-7094-91-06417-3
- 6.
P.
Järvi and M.
Vuorinen, Uniformly perfect sets and quasiregular mappings, J.
London Math. Soc. (2) 54 (1996), no. 3,
515–529. MR 1413895
(98d:30031), http://dx.doi.org/10.1112/jlms/54.3.515
- 7.
Bernard
Maskit, Comparison of hyperbolic and extremal lengths, Ann.
Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381–386.
MR 802500
(87c:30062)
- 8.
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
(98h:30060)
- 9.
C.
McMullen, Iteration on Teichmüller space, Invent. Math.
99 (1990), no. 2, 425–454. MR 1031909
(91a:57008), http://dx.doi.org/10.1007/BF01234427
- 10.
Curtis
T. McMullen, Complex dynamics and renormalization, Annals of
Mathematics Studies, vol. 135, Princeton University Press, Princeton,
NJ, 1994. MR
1312365 (96b:58097)
- 11.
P.
J. Myrberg, Die Kapazität der singulären Menge der
linearen Gruppen, Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys.
1941 (1941), no. 10, 19 (German). MR 0016138
(7,516c)
- 12.
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 0466533
(57 #6411)
- 13.
Ch.
Pommerenke, Uniformly perfect sets and the Poincaré
metric, Arch. Math. (Basel) 32 (1979), no. 2,
192–199. MR
534933 (80j:30073), http://dx.doi.org/10.1007/BF01238490
- 14.
Ch.
Pommerenke, On uniformly perfect sets and Fuchsian groups,
Analysis 4 (1984), no. 3-4, 299–321. MR 780609
(86e:30044)
- 15.
P. Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces, J. Differential Geom. 52 (1999), 407-452. CMP 2000:13
- 16.
R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc. 128 (2000), 2569-2575. CMP 2000:44
- 17.
T. Sugawa, On the geometry of hyperbolic
 -orbifolds, preprint
- 18.
Toshiyuki
Sugawa, Various domain constants related to uniform
perfectness, Complex Variables Theory Appl. 36
(1998), no. 4, 311–345. MR 1670075
(99h:30043)
- 19.
Masatsugu
Tsuji, On the capacity of general Cantor sets, J. Math. Soc.
Japan 5 (1953), 235–252. MR 0058029
(15,309h)
- 20.
M.
Tsuji, Potential theory in modern function theory, Maruzen Co.
Ltd., Tokyo, 1959. MR 0114894
(22 #5712)
- 21.
Pekka
Tukia, On limit sets of geometrically finite Kleinian groups,
Math. Scand. 57 (1985), no. 1, 29–43. MR 815428
(87d:30045)
- 1.
- L. V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand, 1966. MR 34:336
- 2.
- A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), 475-483. MR 80a:30020
- 3.
- L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups: I, Ann. of Math. (2) 91 (1970), 570-600. MR 45:7044
- 4.
- C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), 1-39. MR 98k:22043
- 5.
- R. D. Canary, The Poincaré metric and a conformal version of a theorem of Thurston, Duke Math. J. 64 (1991), 349-359. MR 92k:57020
- 6.
- P. Järvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. 54 (1996), 515-529. MR 98d:30031
- 7.
- B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381-386. MR 87c:30062
- 8.
- K. Matsuzaki, Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces, Geometric Complex Analysis (J. Noguchi et al., ed.), World Scientific, Singapore, 1996, pp. 443-450. MR 98h:30060
- 9.
- C. McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990), 425-454. MR 91a:57008
- 10.
- -, Complex Dynamics and Renormalization, Ann. of Math. Studies, Princeton, 1994. MR 96b:58097
- 11.
- P. J. Myrberg, Die Kapazität der singulären Menge der linearen Gruppen, Ann. Acad. Sci. Fenn. A I Math.-Phys. 10 (1941), 1-19. MR 7:516c
- 12.
- D. Niebur and M. Sheingorn, Characterization of Fuchsian groups whose integrable forms are bounded, Ann. of Math. (2) 106 (1977), 239-258. MR 57:6411
- 13.
- Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (1979), 192-199. MR 80j:30073
- 14.
- -, On uniformly perfect sets and Fuchsian groups, Analysis 4 (1984), 299-321. MR 86e:30044
- 15.
- P. Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces, J. Differential Geom. 52 (1999), 407-452. CMP 2000:13
- 16.
- R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc. 128 (2000), 2569-2575. CMP 2000:44
- 17.
- T. Sugawa, On the geometry of hyperbolic -orbifolds, preprint
- 18.
- -, Various domain constants related to uniform perfectness, Complex Variables Theory Appl. 36 (1998), 311-345. MR 99h:30043
- 19.
- M. Tsuji, On the capacity of general Cantor sets, J. Math. Soc. Japan 5 (1953), 235-252. MR 15:309h
- 20.
- -, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. MR 22:5712
- 21.
- P. Tukia, On limit sets of geometrically finite groups, Math. Scand. 57 (1985), 29-43. MR 87d:30045
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Additional Information
Toshiyuki Sugawa
Affiliation:
Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan
Address at time of publication:
Department of Mathematics, University of Helsinki, P. O. Box 4 (Yliopistonkatu 5), FIN-00014, Helsinki, Finland
Email:
sugawa@kusm.kyoto-u.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002-9947-01-02775-1
PII:
S 0002-9947(01)02775-1
Keywords:
Uniformly perfect,
Kleinian group,
translation length,
Hausdorff dimension
Received by editor(s):
June 16, 1998
Received by editor(s) in revised form:
November 27, 2000
Posted:
May 4, 2001
Dedicated:
Dedicated to Professor Hiroki Sato on the occasion of his sixtieth birthday.
Article copyright:
© Copyright 2001 American Mathematical Society
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