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Subgroups of word hyperbolic groups in rational dimension 2 · arXivDesk

1811.09220Nov 22, 2018First published in: Arora Shivam, Martinez Pedroza Eduardo, SUBGROUPS OF WORD HYPERBOLIC GROUPS IN RATIONAL DIMENSION 2. Groups Geom. Dyn

Subgroups of word hyperbolic groups in rational dimension 2

Shivam Arora, Eduardo Martínez-Pedroza

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 $\mathsf{cd}_{\mathbb{Q}}(G)=2$

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. In particular, our result applies to the class of torsion-free hyperbolic groups

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with

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and

\mathsf{cd}_{\mathbb{Q}}(G)=2

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