10.46298/jgcc.2020.12.1.6059
Delgado, Jordi
Jordi
Delgado
Silva, Pedro V.
Pedro V.
Silva
On the lattice of subgroups of a free group: complements and rank
episciences.org
2020
Mathematics - Group Theory
contact@episciences.org
episciences.org
2020-01-28T18:15:32+01:00
2020-03-02T12:56:10+01:00
2020-03-02
eng
Journal article
https://gcc.episciences.org/6059
arXiv:1905.12597
1869-6104
PDF
1
journal of Groups, complexity, cryptology ; Volume 12, Issue 1 ; 1869-6104
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.
Comment: 27 pages, 5 figures