On the torsion in a group $\mathbf{F}\mathbf{/}[\mathbf{M},\mathbf{N}]$ in the case of combinatorial asphericity of groups $\mathbf{F}\mathbf{/}\mathbf{M}$ and $\mathbf{F}\mathbf{/}\mathbf{N}$

Let $F$ be a non-Abelian free group with basis $A$, $M$ and $N$ be the normal closures of sets $R_M$ and $R_N$ of words in the alphabet $A^{\pm 1}$. As is known, the group $F\mathrm{/}[N,N]$ is torsion-free, but, in general, torsion in $F\mathrm{/}[M,N]$ is possible. In the paper of Hartley and Kuz’min (1991), it was proved that if $R_M=\{v\}$, $R_N=\{w\}$ and words $v$ and $w$ are not a proper power in $F$, then $F\mathrm{/}[M,N]$ is torsion-free. In the present paper a sufficient condition for the absence of torsion in $F\mathrm{/}[M,N]$ is obtained, which allows to generalize the result of Hartley and Kuz’min to arbitrary words $v$ and $w$.