On relatively aspherical presentations and their central extensions

Under the condition of asphericity of a quotient group $G/\bar N_R$, mutual commutants of the form $[\bar N_R, G]$ in hyperbolic groups $G$ are investigated together with the structure of central subgroups $\bar N_R/[\bar N_R, G]$ in central extensions $G/[\bar N_R, G]$ of $G/\bar N_R$. In particular, quotients of the form $G/[g^m,G]$ are considered, where $g$ is an element of infinite order from a hyperbolic group $G$ and $m$ is sufficiently large (depending on $g$).