Exponential groups 2: Extensions of centralizers and tensor completion of CSA groups

For a CSA group $G$ and a wide class of abelian groups $A$ we give an explicit construction for the tensor $A$-completion of $G$ using free products with amalgamations. We apply the obtained results to the study of basic properties of $A$-free groups. In particular, canonical and reduced forms of elements in $A$-free groups are introduced, and then commuting and conjugate elements are described.