Conical Limit Points and the Cannon-thurston Map

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 : ∂G → 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 CannonThurston 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 ∈ 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 ∈ ∂G such that x is not a “controlled concentration point” for the action of G on ∂G.