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

## TL;DR

This paper investigates the structure of subgroups within free groups, focusing on complements and ranks, using automata theory to characterize their properties and relationships.

## Contribution

It introduces new characterizations of complements in free groups, computes their minimal ranks, and explores the conditions under which different types of complements coincide.

## Key findings

- Characterization of when complements exist
- Calculation of the vf3- and f3-cork of subgroups
- Language-theoretical description of cyclic complements

## Abstract

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.