MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

3 added 274 characters in body

If $\langle X,R \rangle$ is a finite presentation of a group $G$, then there exists an exact sequence of $\mathbb ZG$-modules $$0 \to \pi_2(Z) \to \mathbb Z G^{\oplus R} \to \mathbb Z G^{\oplus X} \to \mathbb Z G \to \mathbb Z \to 0,$$ where $Z$ is the presentation $2$-complex of the presentation above. If one knows in addition that $G$ is of type $FP_3$, then $\pi_2(Z)$ must be finitely generated as a $\mathbb Z G$-module. It is well-known that hyperbolic groups are $FP_{\infty}$, using the Rips complex.

Without any assumptions it probably fails

Any example of a finitely presented group which is not of type $FP_3$ gives a counterexample, even though I could i.e. $\pi_2$ is not find finitely generated. Brady constructed a good examplesubgroup of a hyperbolic group with this property in

Brady, N. Branched Coverings of Cubical Complexes and Subgroups of Hyperbolic Groups J. London Math. Soc. (1999) 60(2): 461-480.

Much earlier, Stallings gave an example where the third homology is not finitely generated as a module over the group ring of a finitely presented group.

Stallings, J. A finitely presented group whose 3-dimensional integral homology is not finitely generated. Amer. J. Math. 85 (1963), 541–543.

2 added 375 characters in body

If $\langle X,R \rangle$ is a finite presentation of a group $G$, then there exists an exact sequence of $\mathbb ZG$-modules $$0 \to \pi_2(Z) \to \mathbb Z G^{\oplus R} \to \mathbb Z G^{\oplus X} \to \mathbb Z G \to \mathbb Z \to 0,$$ where $Z$ is the presentation $2$-complex of the presentation above. If one knows in addition that $G$ is of type $FP_3$, then $\pi_2(Z)$ must be finitely generated as a $\mathbb Z G$-module. It is well-known that hyperbolic groups are $FP_{\infty}$, using the Rips complex.

Without any assumptions it probably fails, even though I could not find a good example. Stallings gave an example where the third homology is not finitely generated as a module over the group ring of a finitely presented group.

Stallings, J. A finitely presented group whose 3-dimensional integral homology is not finitely generated. Amer. J. Math. 85 (1963), 541–543.

Post Undeleted by Andreas Thom
Post Deleted by Andreas Thom
1