On independent families of normal subgroups in free groups
Consider a presentation $\mathcal{P}=<{\bf x}\mid{\bf \bigcup_{i=1}^n r_i}>$. Let ${\bf R_i}$ be the normal closure of the set ${\bf r_i}$ in the free group ${\bf F}$ with basis ${\bf x}$, $\mathcal{P}_i=<{\bf x}\mid{\bf r_i}>$, ${\bf N_i} = \prod_{j\neq i}{\bf R_j}$. In the present article, using geometric techniques of pictures, generators for $\frac{{\bf R_i}\cap {\bf N_i}}{[{\bf R_i}, {\bf N_i}]}$, $i=1,...,n$, are obtained from a set of generators over $\{\mathcal{P}_i\mid i=1,..., n\}$ for $\pi_2(\mathcal{P})$. As a corollary, we get a sufficient condition for the family $\{{\bf R_1},...,{\bf R_n}\}$ to be independent.