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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.

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