On Independent Families of Normal Subgroups in Free Groups

Abstract

Consider a presentation \( \mathcal{P}=\left\langle \mathrm{X}\left|\underset{i=1}{\overset{n}{\cup }}{\mathrm{r}}_i\right.\right\rangle \) . Let R_i be the normal closure of the set r_i in the free group F with basis x, \( {\mathcal{P}}_i=\left\langle \mathrm{X}\left|{\mathrm{r}}_i\right.\right\rangle, {\mathrm{N}}_i=\prod \limits_{j\ne i}{\mathbf{R}}_j \). In this paper, using geometric techniques of pictures, generators for \( \frac{{\mathbf{R}}_i\cap {\mathbf{N}}_i}{\left[{\mathbf{R}}_i,{\mathbf{N}}_i\right]},i=1 \), . . . , n, are obtained from a set of generators over \( \left\{{\mathcal{P}}_i\left|i=1\right.,\dots n\right\} \) for \( {\pi}_2\left(\mathcal{P}\right) \). As a corollary, we get a sufficient condition for the family {R₁,…,R_n} to be independent.