Relative hyperbolicity, trees of spaces and Cannon-Thurston maps

Abstract

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 generalizes a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.

Ending Laminations and Cannon–Thurston Maps

“…Let J denote the collection of copies of J obtained in this construction and let (PEY, d pel ) denote the resulting partially electrocuted space. (See [MP11] for a more general discussion.) We have the following basic Lemma.…”

Conical limit points and the Cannon-Thurston map

“…where K ≤ G is a quasiconvex free subgroup of rank 2 and where K 1 ≤ K is also free of rank 2 (and hence K 1 is also quasiconvex in G). Therefore, by a general result of Mitra [69] (see also [78]) about graphs of groups with hyperbolic edge and vertex groups, there does exist a Cannon-Thurston map i : ∂G → ∂G * . Since G ≤ G * is not quasiconvex, Proposition 2.13 implies that the map i is not injective.…”

Cannon–thurston Maps

“…Conversely, ifî(p) =î(q) for p = q ∈ ∂Γ H , then some bi-infinite geodesic having p, q as its end-points is a leaf of Λ EL . [MP11]. Let X and Y be relatively hyperbolic spaces, hyperbolic relative to the collections H X and H Y of 'horosphere-like sets' respectively.…”