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 boolean OR {x(1)(+/- 1),..., x(n)(+/- 1)}. It is proved that, for n >= 2, the group (G) over tilde = < G, x(1), x(2),..., x(n) | w = 1 > always contains a non-Abelian free subgroup. For n = 1, the question whether there exist non-Abelian free subgroups in (G) over tildeG is amply settled for the unimodular case (i.e., where the exponent sum of x(1) in w is one). Some generalizations of these results are discussed.