Relatively hyperbolic extensions of groups and Cannon-Thurston maps

Abstract

Let 1 → (K, K1) → (G, NG(K1)) → (Q, Q1) → 1 be a short exact sequence of pairs of finitely generated groups with K1 a proper non-trivial subgroup of K and K strongly hyperbolic relative to K1. Assuming that, for all g ∈ G, there exists kg ∈ K such that gK1g−1 = kgK1kg−1, we will prove that there exists a quasi-isometric section s: Q → G. Further, we will prove that if G is strongly hyperbolic relative to the normalizer subgroup NG(K1) and weakly hyperbolic relative to K1, then there exists a Cannon-Thurston map for the inclusion i: ΓK → ΓG.