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Equivariant, almost homeomorphic maps between $$S^ 1$$ and $$S^ 2$$. (English) Zbl 0855.57012
Summary: Let $$\Pi$$ be a Fuchsian group isomorphic to a nontrivial, 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, J. Cannon, and W. Thurston [Group invariant Peano curves (preprint)] and Y. N. Minsky [J. Am. Math. Soc. 7, 539-588 (1994; Zbl 0808.30027)] showed that there exists a $$\rho$$-equivariant, surjective, continuous map $$f: S^1_\infty\to S^2_\infty$$. In this paper, we prove that there exist zero-measure sets $$\Lambda^1$$ in $$S^1_\infty$$ and $$\Lambda^2$$ in $$S^2_\infty$$ such that the restriction $$f|_{S^1_\infty- \Lambda^1}: S^1_\infty- \Lambda^1\to S^2_\infty- \Lambda^2$$ is a homeomorphism.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57M60 Group actions on manifolds and cell complexes in low dimensions
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