Free subgroups of one-relator relative presentations

Suppose that G is a non-trivial torsion-free group and w is a word over the alphabet G ⋃ {x ₁ ^±1 ,⋯,x _n ^±1 }. It is proved that, for n ⩾ 2, the group always contains a non-Abelian free subgroup. For n = 1, the question whether there exist non-Abelian free subgroups in is amply settled for the unimodular case (i.e., where the exponent sum of x₁ in w is one). Some generalizations of these results are discussed. Supported by RFBR grant No. 05-01-00895. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 290–298, May–June, 2007.