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, the 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.