Propagating quasiconvexity from fibers

TL;DR

This paper investigates how quasiconvexity properties in hyperbolic groups are preserved and propagated through group extensions, especially focusing on subgroups related to fibers in exact sequences.

Contribution

It establishes conditions under which quasiconvexity of subgroups in fiber groups implies quasiconvexity in the ambient hyperbolic group, extending known results to various group constructions.

Findings

Quasiconvexity propagates from fibers to the entire group under certain conditions.

Results apply to metric bundles, complexes of groups, and graphs of groups.

Provides new insights into subgroup structure in hyperbolic group extensions.

Abstract

Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ 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.