Extensions of groups (Ex)

Are there extensions of groups $1 \to H \to G \to \Z$$<wikitex>; Are there extensions of groups 1 \to H \to G \to \Z$ such that $G$$G$ is hyperbolic and $H$$H$ is nontrivial, torsionfree, and finitely generated?

\to H \to G \to \Z$ such that $G$ is hyperbolic and $H$ is nontrivial, torsionfree, and finitely generated? [[Category:Exercises]] [[Category:Exercises without solution]]1 \to H \to G \to \Z such that $G$$G$ is hyperbolic and $H$$H$ is nontrivial, torsionfree, and finitely generated?