On a class of poly-context-free groups generated by automata

TL;DR

This paper explores the algebraic structure of automaton groups generated by trees, introduces a broad class of reducible automaton groups, and establishes their properties, including their relation to poly-context-free groups and amenability.

Contribution

It characterizes the semigroup of tree automaton groups, introduces reducible automaton groups, and proves their structure as direct limits of poly-context-free groups, supporting a conjecture by T. Brough.

Findings

Semigroup is isomorphic to a partially commutative monoid.

Tree automaton groups with ≥2 generators are not finitely presented.

These groups are amenable and are direct limits of non-amenable groups.

Abstract

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is isomorphic to the partially commutative monoid associated with the complement of the line graph of the defining tree. After that, we generalize these groups by introducing the quite broad class of reducible automaton groups, which lies in the class of contracting automaton groups without singular points. We give a general structure theorem that shows that all reducible automaton groups are direct limit of poly-context-free groups which are virtually subgroups of the direct product of free groups; notice that this result partially supports a conjecture by T. Brough. Moreover, we prove that tree automaton groups with at least two generators are not finitely presented and they are amenable groups, which are direct limit of non-amenable groups.