Is there a good reference for information about the second homotopy group of the Cayley complex or [Presentation complex](http://en.wikipedia.org/wiki/Presentation_complex) of a finitely presented group, especially a hyperbolic group? I'm looking for an argument that the second homotopy group of the Cayley complex of a hyperbolic group $G$ is finitely generated as a $G$-module in particular, but I'd welcome other interesting starting points around the second homotopy group of the Cayley complex too.



Second homotopy group of Cayley complex
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.


Any example of a finitely presented group which is not of type $FP\_3$ gives a counterexample, i.e. $\pi\_2$ is not finitely generated. Brady constructed a subgroup 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.


