On the existence of a free subgroup of finite index

Let G be a finitely generated accessible group. We will study the homology of G with coefficients in the left G-module H¹(G;$\text{Z}$G]). This G-module may be identified with the G-module of continuous functions with values in $\text{Z}$ on the G-space of ends of G, quotiented by the constant functions. The main result is as follows: Suppose G is infinite, then the abelian group H₁(G;H¹(G;$\text{Z}$G])) has rank 1 if G has a free subgroup of finite index and it has rank 0 if G has not.