Self-similar groups acting essentially freely on the boundary of the binary rooted tree

TL;DR

This paper investigates groups generated by automata acting freely on the boundary of a rooted tree, providing tools for classification and a complete analysis of 3-state automata over a 2-letter alphabet.

Contribution

It introduces methods to identify such groups and offers the first comprehensive classification for a specific automaton class.

Findings

Established criteria for essential freeness of automaton groups

Connected these groups to just-infinite and scale-invariant groups

Classified all 3-state automaton groups over a 2-letter alphabet in this class

Abstract

We study the class of groups generated by automata that act essentially freely on the boundary of a rooted tree. In the process we establish and discuss some general tools for determining if a group belongs to this class, and explore the connections of this class to the classes of just-infinite and scale-invariant groups. Our main application is a complete classification of groups generated by 3-state automata over 2-letter alphabet that are in this class.