On Almost-Dense Extension Groups of Torsion-Free Groups

LetAbe a torsion-free group. A mixed groupGis said to be an almost dense extension group (ADE group) ofAifAis almost dense inGandT(G)-high ofG. In the case thatAis of rank 1 andGpis cyclic for every primep, we establish the structure theorem, the representation theorem, the realization theorem, and the classification theorem. Moreover, for such an ADE groupG, we give a necessary and sufficient condition thatGis splitting and establish a characterization that allT(G)-high subgroups ofGare isomorphic.