Cannon-Thurston Maps and Relative Hyperbolicity.

Let P : Y → T be a tree of strongly relatively hyperbolic spaces such that Y is also a strongly relatively hyperbolic space. Let X be a vertex space and i : X ֒→ Y denote the inclusion. The main aim of this thesis is to extend i to a continuous map i : X → Y , where X and Y are the Gromov compactifications of X and Y respectively. Such continuous extensions are called Cannon-Thurston maps. This is a generalization of [Mit98b] which proves the existence of Cannon-Thurston maps for X and Y hyperbolic. By generalizing a result of Mosher [Mos96], we will also prove the existence of a Cannon-Thurston map for the inclusion of a strongly relatively hyperbolic normal subgroup into a strongly relatively hyperbolic group. Let us first briefly sketch the genesis of this problem.Let H be an infinite quasi-convex subgroup of a word hyperbolic group G. We choose a finite generating set of G that contains a finite generating set of H. Let ΓH, ΓG be their respective Cayley graphs with respect to these finite generating sets. Let ∂ΓH and ∂ΓG be hyperbolic boundaries of ΓH and ΓG respectively. Then it is easy to show that the inclusion i: ΓH → ΓG canonically extends to a continuous map from ΓH ∪ ∂ΓH to ΓG ∪ ∂ΓG. But if H is not quasi-convex, it is not clear whether there is such an extension. It turns out that for a wide class of non-quasiconvex subgroups such an extension is possible. The first example of this sort was given by J.Cannon and W.Thurston in [CT07] (1989). They showed that if G is the fundamental group of a closed hyperbolic 3-manifold M fibering over a circle with fiber a closed surface S and if H is the fundamental group of S, then there exists a continuous extension for the embedding i: ΓH → ΓG. In [Min94], Y.N.Minsky generalized Cannon-Thurston’s result to bounded geometry surface Kleinian groups without parabolics. Later on, Mitra, in [Mit98a, Mit98b] (1998), gave a different proof of Cannon-Thurston’s original result and generalized it in the following two directions:Theorem 0.0.1. (Mitra [Mit98a]) Let G be a hyperbolic group and let H be a hyperbolic subgroup that is normal in G. Let i: ΓH → ΓG denote the inclusion. Then i extends to a continuous map ˜i: ΓH ∪ ∂ΓH → ΓG ∪ ∂ΓG.Theorem 0.0.2. (Mitra [Mit98b]) Let (X, d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T. If X is hyperbolic then there exists a Cannon-Thurston map for i: Xv → X, where Xv is the vertex space corresponding to v.Let Σ be a compact surface of genus g(Σ) ≥ 1 with a finite non-empty collection of boundary components {C1, ..., Cm}. Subgroups of π1(Σ) corresponding to the fundamental groups of the boundary curves are called peripheral subgroups. Consider a discrete and faithful action of π1(Σ) on H3 . The action is strictly type preserving if the maximal parabolic subgroups are precisely the peripheral subgroups of π1(Σ). Let N be the quotient manifold obtained from H3 under this action.