Propagating quasiconvexity from fibers

Abstract

Let $1\to K⟶G\stackrel{\pi }{⟶}Q$ be an exact sequence of hyperbolic groups. Let $Q_1<Q$ be a quasiconvex subgroup and let $G_1={\pi }^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed surface group or a free group and $Q$ is convex cocompact), we show that infinite index quasiconvex subgroups of $G_1$ are quasiconvex in $G$. Related results are proven for metric bundles, developable complexes of groups, and graphs of groups.

final version, 21 pages