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Local connectivity, Kleinian groups and geodesics on the blowup of the torus. (English) Zbl 1061.37025
Let \(N = {\mathbb H}^{3}/\Gamma\) be a complete hyperbolic 3-manifold with a free fundamental group \(\pi_{1}(N) \cong \Gamma \cong \langle A, B \rangle,\) such that the commutator \([A, B]\) is parabolic. The limit set \(\Lambda\) of \(N\) is the locus of chaotic dynamics for the action of \(\pi_{1}(N)\) on \(S^{2}_{\infty} = \partial{\mathbb H}^{3}.\) The author shows that the topological dynamical system \((\Lambda, \pi_{1}(N))\) is always a quotient of the standard action of a surface group on a circle. Let \(\Sigma\) be a compact surface of genus one with a single boundary component. Its interior \(\Sigma^0\) can be endowed with a complete hyperbolic metric of finite volume, providing a natural action of \(\pi_{1}(\Sigma)\) on the circle \(S^{1}_{\infty} = \partial\widetilde{\Sigma}^{0} \cong \partial{\mathbb H}.\) There is a homotopy equivalence or marking \(f : \Sigma \rightarrow N, \) sending \(\partial\Sigma\) to a cusp of \(N.\) Let \(H(\Sigma)\) denote the set of all such marked hyperbolic 3-manifolds.
Some results are the following theorem and corollary: Theorem 1.1. For any \(N \in H(\Sigma)\) there is a natural, continuous, surjective map \(F : S^{1}_{\infty} \rightarrow \Lambda \subset S^{2}_{\infty},\) respecting the action of \(\pi_{1}(\Sigma).\) Corollary 1.2. The limit set of any \(N \in H(\Sigma)\) is locally connected. Conjecture 1.3. For any hyperbolic 3-manifold \(N\) with finitely generated fundamental group, there exists a continuous, \(\pi_{1}(N)\)-equivariant map \(F : \partial \pi^{1}(N) \rightarrow \Lambda \subset S^{2}_{\infty}.\) Theorem 1.1 is a special case of this conjecture.

MSC:
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F60 Teichmüller theory for Riemann surfaces
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