On independent families of normal subgroups in free groups

Abstract

Consider a presentation $\mathcal{P}=<\mathbf{x}\mid {\bigcup }_{\mathbf{i}=1}^{\mathbf{n}}{\mathbf{r}}_{\mathbf{i}}>$. Let ${\mathbf{R}}_{\mathbf{i}}$ be the normal closure of the set ${\mathbf{r}}_{\mathbf{i}}$ in the free group $\mathbf{F}$ with basis $\mathbf{x}$, ${\mathcal{P}}_i=<\mathbf{x}\mid {\mathbf{r}}_{\mathbf{i}}>$, ${\mathbf{N}}_{\mathbf{i}}={\prod }_{j\ne i}{\mathbf{R}}_{\mathbf{j}}$. In the present article, using geometric techniques of pictures, generators for $\frac{{\mathbf{R}}_{\mathbf{i}}\cap {\mathbf{N}}_{\mathbf{i}}}{[{\mathbf{R}}_{\mathbf{i}},{\mathbf{N}}_{\mathbf{i}}]}$, $i=1,...,n$, 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 $\{{\mathbf{R}}_1,...,{\mathbf{R}}_{\mathbf{n}}\}$ to be independent.

13 pages