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Conical limit points and the Cannon-Thurston map. (English) Zbl 1375.20045
Summary: Let \( G\) be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space \( Z\) so that there exists a continuous \( G\)-equivariant map \( i:\partial G\to Z\), which we call a Cannon-Thurston map. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in \( Z\) in terms of their pre-images under the Cannon-Thurston map \( i\). As an application we prove, under the extra assumption that the action of \( G\) on \( Z\) has no accidental parabolics, that if the map \( i\) is not injective, then there exists a non-conical limit point \( z\in Z\) with \( | i^{-1}(z)|=1\). This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if \( G\) is a non-elementary torsion-free word-hyperbolic group, then there exists \( x\in \partial G\) such that \( x\) is not a “controlled concentration point” for the action of \( G\) on \( \partial G\).

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
20F65 Geometric group theory
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37F40 Geometric limits in holomorphic dynamics
37F30 Quasiconformal methods and Teichm├╝ller theory, etc. (dynamical systems) (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
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