On the lattice of subgroups of a free group: complements and rank (Q6566687)

Let \(F_{n}\) be a free group of rank \(n\) and let \(H \leq F_{n}\) be a finitely generated subgroup of \(F_{n}\). A \(\vee\)-complement of \(H\) is a subgroup \(K \leq F_{n}\) such that \(H \vee K =\langle H, K \rangle=F_{n}\), if \(H \cap K=1\), then \(K\) is called a \(\oplus\)-complement. The minimum possible rank of a \(\vee\) complement (resp., \(\oplus\)-complement) of \(H\) is called the \(\vee\)-orank (resp., \(\oplus\)-corank) of \(H\).\N\NThe authors use Stallings automata (see [\textit{J. R. Stallings}, Invent. Math. 71, 551--565 (1983; Zbl 0521.20013)]) to study these notions and the relations between them. In particular, they characterize when complements exist, compute the \(\vee\)-corank and provide language-theoretical descriptions of the sets of cyclic complements. Finally, they prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds.