Abstract
Let \(F\) be a nonabelian free group with basis \(A\) , and let \(M\) and \(N\) be the normal closures of sets \(R_M\) and \(R_N\) of words in the alphabet \(A^{\pm 1}\) . As is known, there is no torsion in the group \(F/[N,N]\) ; however, in general, a torsion in \(F/[M, N]\) is possible. In the paper by Kuz’min and Hartley (1991), it was proved that if \(R_M=\{v\}\) , \(R_N=\{w\}\) , and the words \(v\) and \(w\) are not proper powers in \(F\) , then there is no torsion in \(F/[M,N]\) . In this paper, we obtain a sufficient condition for the absence of torsion in \(F/[M,N]\) , which enables us to generalize the result of Kuz’min and Hartley to arbitrary words \(v\) and \(w\) .
