The SQ-universality of one-relator relative presentations
  Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance, adding one generator and one relator in which the exponent sum of the additional generator is one to a free product of two nontrivial torsion-free groups, we also obtain an SQ-universal group.   Key words: relative presentations, one-relator groups, SQ-universality, equations over groups. 
Anton A. Klyachko