Question

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.

Answer 1

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.

Answer 2

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 2-complexes. 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.

Answer 3

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 Low-dimensional topology (Bangor, 1979), London Math. Soc. Lecture Note Ser., Volume~48. Cambridge Univ. Press, Cambridge (1982), 153--202.

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) 28--61 (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 2-complex $K$, $\pi_2(K)$ 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.