A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela and provides a purely topological proof that f...

In this chapter we discuss cyclic groups, the quotient group construction, the direct sum construction and the first isomorphism theorem, in the context of additive abelian groups; we also discuss free modules. These concepts are necessary, as well as the matrix theory of Chapter 1, for the study of finitely generated abelian groups in Chapter 3. At the same time the material provides the reade...

If G is a finite abelian group, R is a principal ideal domain with field of quotients an algebraic number field K which splits G, and if A is a Hopf algebra order in KG, then the Grothendieck ring of the category of finitely generated A-modules is isomorphic to the Grothendieck ring of the category of finitely generated RG-modules. The Grothendieck group a£ of the category of finitely generated...

Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support |Σ| of a fan Σ. A unimodular fan ∆ over |Σ| determines a Schauder basis ofG: its elements are the minimal positive free generators of the pointwise ordered group of...