Free subgroups of one-relator relative presentations

Abstract

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group G=<G,x_1,x_2,...,x_n | w=1> always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.