Faithful Group Actions and Aspherical Complexes

For a free group $F$ and normal subgroups $R$ and $S$ of $F$, we study the question of whether the action of the group $F/RS$ on the abelian group $\frac {R\cap S}{[R,S]}$ with respect to conjugation in $F$ is faithful. We find conditions on the subgroups $R$ and $S$ under which this action is faithful and apply this theory to the study of the asphericity of two-dimensional CW-complexes and derived series in groups. One of the applications of the method considered in this paper is a description of obstructions to the asphericity of the so-called LOT presentations in terms of transfinite derived series.