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

A ∨-complement of a subgroup H ⩽ Fn is a subgroup K ⩽ Fn such that H ∨ K = Fn. If we also ask K to have trivial intersection with H, then we say that K is a ⊕-complement of H. The minimum possible rank of a ∨-complement (resp. ⊕-complement) of H is called the ∨-corank (resp. ⊕-corank) of H.

We use Stallings automata to study these notions and the relations between them. In particular, we characterize when complements exist, compute the ∨-corank, and provide language-theoretical descriptions of the sets of cyclic complements. Finally, we prove that the two notions of corank coincide on subgroups that admit cyclic complements of both kinds.

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