# zbMATH — the first resource for mathematics

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
Full Text: