Classes of free group extensions (Q6142859)

Let \(F=F_{B}\) be a free group with basis \(B\). In [\textit{J. R Stallings}, Invent. Math. 71, 551--565 (1983; Zbl 0521.20013)] is established a correspondence between subgroups of \(F\) with labeled graphs called core graphs. To each subgroup \(H\) of \(F\) is associated a pointed labeled graph \(\Gamma_{B}(H)\) and for subgroups \(K \leq H \leq F\) a graph morphism \(\Gamma_{B}(H) \rightarrow \Gamma_{B} (K)\). Thus, the category of subgroups of free groups ordered by inclusion is a subcategory of the category of pointed labeled graphs. This article is a development of the author's paper [J. Algebra 569, 595--615 (2021; Zbl 1477.20048)]. Theorem 3.2 states that if \(H \leq K \leq F_{Y}\) are finitely generated groups, then there is a basis \(B\) of \(F_{Y}\) such that the associated graph morphism \(\Gamma_{B}(H) \rightarrow \Gamma_{B} (K)\) is onto. Moreover \(B\) can be obtained from \(Y\) by conjugation. In the category of pointed labeled core graphs there are no free group extension such that for every automorphism the graph morphism is injective. The author shows that if graphs without base points are taken into consideration, then there is an extension \[ \langle b \rangle \leq \langle b, b^{a} \rangle \leq F_{\{a,b\}} \] such that for every basis of \(F_{\{a,b\}}\) the associated graph morphisms are injective.