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

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

Blog Article

A $ee$-complement of a subgroup $H leqslant mathbb{F}_n$ is a subgroup Custom Tote Bag $K leqslant mathbb{F}_n$ such that $H ee K = mathbb{F}_n$.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 $ee$-complement (resp.

$oplus$-complement) of $H$ is called the $ee$-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 $ee$-corank, and provide language-theoretical descriptions of the sets of cyclic complements.Finally, we prove that the two notions Truck Model Kit of corank coincide on subgroups that admit cyclic complements of both kinds.