$SQ$-universality of one-relator relative presentations

Adding two generators and an arbitrary relator to a non-trivial torsion-free group one always obtains an $SQ$-universal group. In the course of the proof of this theorem one obtains several other results of independent interest. For instance, the addition of a generator and one relator in which the sum of the exponents of the additional generator is equal to 1 to a free product of two non-trivial torsion-free groups also produces an $SQ$-universal group.
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