On intersections of normal subgroups in free groups

Let $N_1$ (respectively $N_2$) be a normal closure of a set $R_1=\{ u_i\}$ (respectively $R_2=\{v_j\}$) of cyclically reduced words of the free group $F(A)$. In the paper we consider geometric conditions on $R_1$ and $R_2$ for $N_1\cap N_2=[N_1,N_2]$. In particular, it turns out that if a presentation $<A\,\mid R_1,R_2>$ is aspherical (for example, it satisfies small cancellation conditions $C(p)\& T(q)$ with $1/p+1/q=1/2$), then the equality $N_1\cap N_2=[N_1,N_2]$ holds.