Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps

Abstract

Let $1\to (K,K_1)\to (G,N_G(K_1))\to (Q,Q_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$ such that $g{K}_1{g}^{-1}=k{K}_1{k}^{-1}$, we prove that there exists a quasi-isometric section $s:Q\to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i:{\Gamma }_K\to {\Gamma }_G$.

16 pages, No figures