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

Abstract

A $\vee$-complement of a subgroup $H⩽{\mathbb{F}}_n$ is a subgroup $K⩽{\mathbb{F}}_n$ such that $H\vee K={\mathbb{F}}_n$. If we also ask $K$ to have trivial intersection with $H$, then we say that $K$ is a $\oplus$-complement of $H$. The minimum possible rank of a $\vee$-complement (resp. $\oplus$-complement) of $H$ is called the $\vee$-corank (resp. $\oplus$-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 $\vee$-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.

27 pages, 5 figures

References in corpus (1)

• A Fast Algorithm for Stallings' Folding Process