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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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