Abstract:
Suppose that $G$ is a non-trivial torsion-free group and $w$ is a word over the alphabet $G\cup\{x^{\pm1}_1,\dots,x^{\pm1}_n\}$. It is proved that, for $n\geqslant2$, the group $\widetilde G=\langle G,x_1,x_2,\dots,x_n\,|\,w = 1\rangle$ always contains a non-Abelian free subgroup. For $n=1$, the question whether there exist non-Abelian free subgroups in $\widetilde G$ 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.
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