Conical limit points and the Cannon-Thurston map

Abstract

Abstract. 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 characterzations (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 ∈ 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 wordhyperbolic 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.

Cannon-Thurston fibers for iwip automorphisms of FN

“…In particular, if the Cannon-Thurston map ι : ∂H → ∂G exists, then this map is unique and for any sequence h n ∈ H ∪ ∂H converging to some X ∈ ∂H in the topology of H ∪ ∂H, we have lim n→∞ h n = ι(X) in G ∪ ∂G. It is known, see [30, Proposition 2.12], that if H is a non-elementary word-hyperbolic subgroup of a word-hyperbolic group G, then a map ∂H → ∂G is the Cannon-Thurston map if and only if this map is continuous and H-equivariant.…”

Horospheres in Degenerate 3-Manifolds

“…H is the set of ideal points of lifts of almost minimizing geodesic rays. Theorem 1.3 answers an issue that has come up in works of several authors [Kap95,Ger12,JKLO16] who tried to relate the injective points of the Cannon-Thurston map to the conical limit set. They concluded that the conical limit set is strictly contained in the set of injective points of the Cannon-Thurston map.…”

Trees, dendrites and the Cannon–Thurston map

“…We remark that H and G are hyperbolic groups with H a non-elementary subgroup of G, then if the Cannon-Thurston map ∂i : ∂H → ∂G exists, then for each h ∈ H of infinite order, ∂i(h ∞ ) = lim n→∞ (i(h)) n . Additionally, in this setting where H is non-elementary, if f : ∂H → ∂G is a continuous and H-equivariant map, then f must be the Cannon-Thurston map (Proposition 2.12 [37]).…”

“…Note also that if H is a quasiconvex subgroup of a hyperbolic group G, then the Cannon-Thurston map trivially exists (and is injective on ∂H), since geodesic rays with bounded Hausdorff distance in Γ H will map to quasi-geodesic rays which have bounded Hausdorff distance in Γ G . In fact, it is known that the Cannon-Thurston map ∂i : ∂H → ∂G is injective if and only if H is a quasiconvex subgroup of G (see Proposition 2.13 [37] and Lemma 2.1 [51]).…”