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.

中文翻译：

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有理维2中的单词双曲组的子组

Gersten的结果表明，如果$ G $是一个双曲群，具有完整的同调维数\\ mathsf {cd} _ {\ mathbb {Z}}（G）= 2 $，那么每个有限表示的子群都是双曲的。我们针对有理情况$ \ mathsf {cd} _ {\ mathbb {Q}}（G）= 2 $推广该结果。特别是，该结果适用于无扭转双曲组$ G $的类别，其中$ \ mathsf {cd} _ {\ mathbb Z}（G）= 3 $和$ \ mathsf {cd} _ {\ mathbb Q} （G）= 2 $由Bestvina和Mess发现。