Let A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.
