Classes of free group extensions

Abstract

In this paper we identify different classes of free group extension using core graphs, by further developing machinery from [3]. We show that every free group extension H≤K≤F has a basis B such that the associated pointed graph morphism ΓB(H)→ΓB(H) is onto. But if we examine graphs without base points, there is an extension 〈b〉≤〈b,aba−1〉<F{a,b} such that for every basis of F{a,b} the associated graph morphisms are injective.