  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.
