Classes of free group extensions

TL;DR

This paper classifies free group extensions using core graphs, showing that all have a base with onto morphisms, but some extensions exhibit injective morphisms without base points, revealing structural differences.

Contribution

It introduces a classification of free group extensions via core graphs and analyzes properties of graph morphisms with and without base points.

Findings

Every free group extension has a base with an onto graph morphism.

Some extensions have injective morphisms regardless of base points.

Graph morphism properties depend on the presence or absence of base points.

Abstract

In this paper we identify different classes of free group extension using core graphs. We show that every free group extension $H\leq K\leq F$ has a base $B$ such that the associated pointed graph morphism $\Gamma_{B}\left(H\right)\to\Gamma_{B}\left(H\right)$ is onto. But if we examine graphs without base points, there is an extension $\left\langle b\right\rangle \leq\left\langle b,aba^{-1}\right\rangle <F_{\left\{ a,b\right\} }$ such that for every base of $F_{\left\{ a,b\right\} }$ the associated graph morphisms are injective.