Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

Abstract

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.

27pgs No figs, v3: final version, incorporating referee's comments, to appear in Geometriae Dedicata

References in corpus (2)

• A Combination Theorem for Strong Relative Hyperbolicity
• Cannon-Thurston Maps and Bounded Geometry

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• Cubulating Drilled bundles over graphs
• Poisson boundary of groups acting on real trees
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