Is there a good reference for information about the second homotopy group of the Cayley complex or 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.
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 wellknown 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): 461480. 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 3dimensional integral homology is not finitely generated. Amer. J. Math. 85 (1963), 541–543. 


Here are some starting points, for which I am afraid I have to refer to my own work. The old problem of "Identities among relations" was discussed in Brown, R. and Huebschmann, J. Identities among relations. In Lowdimensional topology (Bangor, 1979), London Math. Soc. Lecture Note Ser., Volume~48. Cambridge Univ. Press, Cambridge (1982), 153202. The problem of calculating $\pi_2$ was further attacked in Brown, R. and Razak~Salleh, A. , "Free crossed resolutions of groups and presentations of modules of identities among relations". LMS J. Comput. Math. 2 (1999) 2861 (electronic). which gives a method for a finite group. This involves constructing inductively a universal cover with a contracting homotopy. The nice point is that constructing a "home for a contracting homotopy" can be made computational. These methods have been developed into a big software system by Graham Ellis: Homological algebra programming . http://hamilton.nuigalway.ie/Hap/www/ . The notion of crossed module is behind this, since for a 2complex $K$, $\pi_2(K)$ is the kernel of the crossed module $\pi_2(K^2,K^1,x) \to \pi_1(K^1,x)$. It seems to be a challenge to link methods already developed for crossed modules (e.g. coproducts, induced crossed modules) into geometric group theory. 


A standard way to get at $\pi_2$ of any space is that it is $H_2$ of the universal cover of the space. This is because higher homotopy groups are invariant under covering maps plus the Hurewicz theorem. Presentation complexes are basically just the connected 2complexes. Its not clear to me that $\pi_2$ should be finitely generated over $\pi_1$. Do you have some reason to think it should be true when the group is hyperbolic? It appears to not be true for general presentation complexes. 

