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Connectedness properties of limit sets. (English) Zbl 0938.20033
Let $$M$$ be a continuum, that is, a connected compact Hausdorff topological space, and let $$\Gamma$$ be a convergence group acting on $$M$$. (Convergence groups were introduced by F. Gehring and G. Martin.) This action is minimal if it has no proper non-empty closed invariant set. A parabolic element of $$\Gamma$$ is an infinite order element with exactly one fixed point. If $$G$$ is a two-ended subgroup of $$\Gamma$$, then $$\eta_\Gamma(G)$$ is the number of ends of the pair $$(\Gamma,G)$$. A loxodromic subgroup $$G$$ of $$\Gamma$$ is one whose limit set $$\Lambda G$$ consists of precisely two points. The author defines also $$\eta_M(G)$$ as the number of connected components of the compact Hausdorff space $$(M\setminus\Lambda G)/G$$.
The main result of this paper is Theorem 1. Let $$\Gamma$$ be a one-ended finitely presented group with no infinite torsion subgroup. Let $$M$$ be a metrisable continuum which admits a minimal convergence action by $$\Gamma$$. Suppose that for any loxodromic subgroup $$G\leq\Gamma$$ with $$\eta_\Gamma(G)>1$$, we have $$\eta_M(G)>1$$. Then, every global cut point of $$M$$ is a parabolic fixed point.
As an application, the author proves Theorem 2. Let $$\Gamma$$ be a relatively hyperbolic group whose boundary $$\partial\Gamma$$ is connected. Suppose that each peripheral subgroup is finitely presented, either one-ended or two-ended, and contains no infinite torsion subgroup. Then, every global cut point of $$\partial\Gamma$$ is a parabolic fixed point.
From Theorem 2, the author obtains Corollary 1. The boundary of a one-ended hyperbolic group has no global cut point. Corollary 2. Suppose that $$\Gamma$$ is a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature. If the limit set $$\Lambda\Gamma$$ is connected, then every global cut point is a parabolic fixed point.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 54D05 Connected and locally connected spaces (general aspects) 54F15 Continua and generalizations
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