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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Fuchsian Groups, Quasiconformal Groups,
and Conical Limit Sets


Authors: Peter W. Jones and Lesley A. Ward
Journal: Trans. Amer. Math. Soc. 352 (2000), 311-362
MSC (1991): Primary 30C62, 30F35, 20H10
DOI: https://doi.org/10.1090/S0002-9947-99-02118-2
Published electronically: September 9, 1999
MathSciNet review: 1458326
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.


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Additional Information

Peter W. Jones
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520–8283
Email: jones@math.yale.edu

Lesley A. Ward
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005–1892, and MSRI, 1000 Centennial Drive, Berkeley, California 94720–5070
Address at time of publication: Department of Mathematics, Harvey Mudd College, Claremont, California 91711
Email: ward@math.hmc.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02118-2
Keywords: Fuchsian groups, quasiconformal groups, limit sets
Received by editor(s): February 1, 1996
Received by editor(s) in revised form: May 13, 1997
Published electronically: September 9, 1999
Additional Notes: Partially supported by NSF grant #DMS-92-13595.
Research at MSRI supported in part by NSF grant #DMS-90-22140.
Article copyright: © Copyright 1999 American Mathematical Society