Geodesic
$SQ$-universality of one-relator relative presentations
Sbornik. Mathematics, Tome 197 (2006) no. 10, pp. 1489-1508

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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. Bibliography: 27 titles. 
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     author = {Ant. A. Klyachko},
     title = {$SQ$-universality of one-relator relative presentations},
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Ant. A. Klyachko. $SQ$-universality of one-relator relative presentations. Sbornik. Mathematics, Tome 197 (2006) no. 10, pp. 1489-1508. http://geodesic.mathdoc.fr/item/SM_2006_197_10_a5/