Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps

Details Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps

2008-01-07

arxiv.org/abs/0801.0933v2

Mathematics

Group Theory

16 pages, No figures

Scientific paper

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 $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\colon Q \to G$. 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 $i\colon\Gamma_K\to \Gamma_G$.

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