Subgroups of word hyperbolic groups in rational dimension 2

Abstract

A result of Gersten states that if $G$ is a hyperbolic group with integral cohomological dimension $\mathsf{cd}_{\mathbb{Z}}(G)=2$ then every finitely presented subgroup is hyperbolic. We generalize this result for the rational case $\mathsf{cd}_{\mathbb{Q}}(G)=2$. In particular, our result applies to the class of torsion-free hyperbolic groups $G$ with $\mathsf{cd}_{\mathbb{Z}}(G)=3$ and $\mathsf{cd}_{\mathbb{Q}}(G)=2$ discovered by Bestvina and Mess.