On certain functorial properties of finitely generated groups

We consider chains of finitely presentable groups with additional properties. These additional properties ensure that every finitely generated group can be written as the colimit of such a chain. We prove various results about these chains, including an isomorphism theorem, and show that the category of these chains can be used to describe the category of finitely generated groups "up to natural isomorphism". The last part is an instance of localisations of categories, and is formulated as an equivalence of categories. Finally, we consider the homology of finitely generated groups. We apply spectral sequences and equivariant homology groups to derive long, exact sequences for the homology of three special classes of finitely generated groups.