Abstract
The purpose of this paper is to investigate the topological and analytical restrictions on a domain D in euclidean n-space R n on which an infinite discrete quasiconformal group can act. We will see that the restrictions are indeed severe, unlike the case of a discrete group of topological or differentiable homeomorphisms.
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References
M. Freedman and R. Skora. Strange actions of groups on spheres ,J. Differential Geometry 25 (1987).
D. Gauld and M. K. Vamanamurthy. Quasiconformal extensions of mappings in n-space, Ann. Acad. Sci. Fenn. Ser. A.I. 3 (1977).
F. W. Gehring. Rings and quasiconformal mappings in space ,Trans. A.M.S. 103 (1962).
F. W. Gehring and G. J. Martin. Discrete quasiconformal groups I, II ,J. London Math. Soc. (3) 55 (1987)
F. W. Gehring and O. Martio. Quasiextremal distance domains and extension of quasiconformal mappings ,J. D’Analyse Math. 45 (1985).
F. W. Gehring and B. G. Osgood. Uniform domains and the quasi-hyperbolic metric, J. D’Analyse Math. 36 (1979).
F. W. Gehring and B. Palka. Quasiconformally homogeneous domains ,J. D’Analyse Math. 30 (1976).
M. Gromov. Hyperbolic manifolds, groups and actions ,in “Annals of Math. Studies”, No. 97, Princeton University Press.
G. J. Martin. Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups ,Ann. Acad. Sci. Fenn. Ser. A.I. 11 (1986).
G. J. Martin and F. W. Gehring. Generalizations of Kleinian groups ,M.S.R.I. Preprint series.
M. H. A. Newman. A theorem on periodic transformations of spaces ,Quarterly J. Math. 2 (1931).
T. B. Rushing. “Topological embeddings”, Academic Press, 1973.
D. Sullivan. The ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions ,in “Annals of Math. Studies”, No. 97, Princeton University Press.
P. Tukia. On two dimensional quasiconformal groups ,Ann. Acad. Sci. Fenn. Ser. A.I. 5 (1980).
P. Tukia. A quasiconformal group not isomorphic to a Möbius group ,Ann. Acad. Sci. Fenn. Ser. A.I. 6 (1981).
P. Tukia. On quasiconformal groups ,to appear, J. d’Analyse.
P. Tukia and J. Väisälä. Quasiconformal approximation and extension ,Ann. Acad. Sci. Fenn. Ser. A.I. 6 (1981)
P. Tukia and J. Väisälä. Quasisymmetric embeddings of metric spaces ,Ann. Acad. Sci. Fenn. Ser. A.I. 5 (1980).
J. Väisälä. “Lectures on n-dimensional quasiconformal mappings”, Lecture notes in Math. 229, Springer Verlag, 1971.
J. H. C. Whitehead. A certain open manifold whose group is unity ,Quarterly J. Math. Ser. (2)6 (1935).
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© 1988 Springer-Verlag New York Inc.
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Martin, G.J. (1988). Quasiconformal Actions on Domains in Space. In: Drasin, D., Earle, C.J., Gehring, F.W., Kra, I., Marden, A. (eds) Holomorphic Functions and Moduli II. Mathematical Sciences Research Institute Publications, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9611-6_8
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DOI: https://doi.org/10.1007/978-1-4613-9611-6_8
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