Conical limit points and the Cannon-Thurston map

Abstract

Let G G be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space Z Z so that there exists a continuous G G -equivariant map i : ∂ G → Z 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 Z in terms of their pre-images under the Cannon-Thurston map i i . As an application we prove, under the extra assumption that the action of G G on Z Z has no accidental parabolics, that if the map i i is not injective, then there exists a non-conical limit point z ∈ Z z\in Z with | i − 1 ( z ) | = 1 |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 G is a non-elementary torsion-free word-hyperbolic group, then there exists x ∈ ∂ G x\in \partial G such that x x is not a “controlled concentration point” for the action of G G on ∂ G \partial G .