Freeness Obstructions for Self-Similar Groups Acting on Rooted Trees

Abstract

We prove two general obstructions to freeness for self-similar groups and derive consequences for automaton groups. First, let $\mathcal A$ be a finite invertible Mealy automaton with a co-accessible identity state. Then every finitely generated subgroup of $G(\mathcal A)$ that is either self-similar and transitive on the first level, or level-transitive on the rooted tree, is cyclic or non-free. This answers a question of Grigorchuk and partially extends Sidki's obstruction for automata of polynomial activity. The methods also yield effective, quantitative, and geometric consequences for relations and level Schreier graphs. Second, we study the relation between freeness and bireversibility. We prove that if a finite reduced reversible invertible Mealy automaton generates a non-abelian free group, then its dual has a bireversible connected component. If the states form a free basis and the action is transitive on the first level, then the automaton itself is bireversible. Finally, over a binary alphabet, the free-basis assumption can also be removed: every finite reduced invertible Mealy automaton generating a non-abelian free group and acting level-transitively is bireversible.