Pullbacks of Metric Bundles and Cannon-Thurston Maps

Abstract. Metric (graph) bundles were defined by Mj and Sardar. In this talk, we discuss the notion of morphisms and pullbacks of metric (graph) bundles. Given a metric (graph) bundle X over a hyperbolic space B and a qi embedding i : A→ B, when X and all the fibers are uniformly (Gromov) hyperbolic and nonelementary, we discuss the hyperbolicity of the pullback Y and the existence of the Cannon-Thurston (CT) map for i* : Y → X, i.e., the continuous boundary extension ∂ὶ* : ∂Y→  ∂X. We also look at an application of this theorem which shows that given a short exact sequence of nonelementary hyperbolic groups 1→ N → G →^π Q→ 1 and a finitely generated qi embedded subgroup Q1 < Q, G1 := π^-1(Q₁) is hyperbolic and the inclusion G1→ G admits a CT map ∂G1→ ∂G. This is part of a joint work with Pranab Sardar.