Onto extensions of free groups (Q6566724)

Let \(A\) be an alphabet of \(r\) letters and let \(F_{A}\) be the free group of rank \(r\). An extension of subgroups \(H\leq K\leq F_{r}\) is called onto when, for every ambient basis \(A'\), the Stallings graph \(\Gamma_{A'}(K)\) is a quotient of \(\Gamma_{A'}(H)\) (see [\textit{J. R. Stallings} [Invent. Math. 71, 551--565 (1983; Zbl 0521.20013)]), and it is called free, whenever some (so, any) basis of \(H\) can be extended to a basis for \(K\). Finally, an extension is called algebraic if \(H\) is not contained in any proper free factor of \(K\). Every algebraic extensions is onto [\textit{M. Takahasi}, Osaka Math. J. 3, 221--225 (1951; Zbl 0044.01106)], but the converse implication does not hold in general, see [\textit{O. Parzanchevski} and \textit{D. Puder}, Math. Proc. Camb. Philos. Soc. 157, No. 1, 1--11 (2014; Zbl 1322.20016)] for \(r=2\) and [\textit{N. M. D. Kolodner}, J. Algebra 640, 477--490 (2024; Zbl 1529.20041)] for all \(r \geq 2\).\N\NIn the paper under review, the authors study properties of extension among free groups (as well as the fully onto variant) and investigate their corresponding closure operators.