Propagating quasiconvexity from fibers

Abstract

Let 1→K→G→πQ be an exact sequence of hyperbolic groups. Let Q1 < Q be a quasiconvex subgroup and let G1 = π−1(Q1). 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 G1 are quasiconvex in G. Related results are proven for metric bundles, developable complexes of groups and graphs of groups.