On independent families of normal subgroups in free groups

TL;DR

This paper investigates the structure of normal subgroups in free groups using geometric picture techniques, providing generators for certain intersections and a condition for independence of subgroup families.

Contribution

It introduces a geometric approach to identify generators of subgroup intersections and establishes a sufficient condition for their independence in free groups.

Findings

Generators for intersections of normal closures are explicitly obtained.

A sufficient condition for the independence of a family of normal subgroups is provided.

Uses geometric picture techniques to analyze subgroup relations.

Abstract

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.