## Delgado, Jordi and Silva, Pedro V. - On the lattice of subgroups of a free group: complements and rank

gcc:6059 - journal of Groups, Complexity, Cryptology, March 2, 2020, Volume 12, issue 1
On the lattice of subgroups of a free group: complements and rank

Authors: Delgado, Jordi and Silva, Pedro V.

A $\vee$-complement of a subgroup $H \leqslant \mathbb{F}_n$ is a subgroup $K \leqslant \mathbb{F}_n$ 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.

Volume: Volume 12, issue 1
Published on: March 2, 2020
Submitted on: January 28, 2020
Keywords: Mathematics - Group Theory