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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Equivariant, almost homeomorphic maps between $ S\sp 1$ and $ S\sp 2$


Author: Teruhiko Soma
Journal: Proc. Amer. Math. Soc. 123 (1995), 2915-2920
MSC: Primary 57M50; Secondary 57M60
MathSciNet review: 1277134
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Abstract: Let $ \Pi $ be a Fuchsian group isomorphic to a non-trivial, closed surface group, and let $ M = {\mathbb{H}^3}/\Gamma $ be a hyperbolic 3-manifold admitting an isomorphism $ \rho :\Pi \to \Gamma $. Under certain assumptions, Cannon-Thurston and Minsky showed that there exists a $ \rho $-equivariant, surjective, continuous map $ f:S_\infty ^1 \to S_\infty ^2$. In this paper, we prove that there exist zero-measure sets $ {\Lambda ^1}$ in $ S_\infty ^1$ and $ {\Lambda ^2}$ in $ S_\infty ^2$ such that the restriction $ f{\vert _{S_\infty ^1 - {\Lambda ^1}}}:S_\infty ^1 - {\Lambda ^1} \to S_\infty ^2 - {\Lambda ^2}$ is a homeomorphism.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1277134-3
PII: S 0002-9939(1995)1277134-3
Keywords: Hyperbolic 3-manifolds, hyperbolic surfaces, equivariant maps, measured foliations
Article copyright: © Copyright 1995 American Mathematical Society