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On Cannon-Thurston maps for relatively hyperbolic groups. (English) Zbl 1307.20038
Summary: O. Baker and T. R. Riley [Forum Math. Sigma 1, Article ID e3 (2013; Zbl 1276.20054)] gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon-Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon-Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

20F67 Hyperbolic groups and nonpositively curved groups
57M60 Group actions on manifolds and cell complexes in low dimensions
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
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