On the transition monoid of the Stallings automaton of a subgroup of a free group

TL;DR

This paper explores the algebraic properties of the transition monoid of Stallings automata for subgroups of free groups, linking these properties to subgroup characteristics and pseudovarieties.

Contribution

It extends previous work by connecting algebraic properties of subgroups with those of their transition monoids, especially within specific pseudovarieties.

Findings

Characterizes when the transition monoid belongs to certain pseudovarieties.

Analyzes properties of normal, malnormal, and cyclonormal subgroups via transition monoids.

Establishes new links between subgroup properties and automaton algebraic structures.

Abstract

Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup $K$ of a free group is pure if and only if the transition monoid $M(K)$ of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of $K$ and algebraic properties of $M(K)$. We mainly focus on the cases where $M(K)$ belongs to the pseudovariety $\overline{\boldsymbol{\mathbf{{H}}}}$ of finite monoids all of whose subgroups lie in a given pseudovariety $\overline{\boldsymbol{\mathbf{{H}}}}$ of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of $F_A$ using the transition monoid of the corresponding Stallings automaton.