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

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 of finite monoids all of whose subgroups lie in a given pseudovariety of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of [Formula: see text] using the transition monoid of the corresponding Stallings automaton.