Orbit automata as a new tool to attack the order problem in automaton groups

TL;DR

This paper introduces orbit automata as a novel method to analyze automaton groups, enabling the detection of infinite order elements where previous techniques were ineffective.

Contribution

The paper presents orbit automata as a new tool for studying automaton groups and demonstrates its effectiveness in identifying elements of infinite order.

Findings

Orbit automata effectively identify infinite order elements in automaton groups.

The method reveals new structural insights into automaton group actions.

Previous methods failed to detect certain infinite order elements, but orbit automata succeed.

Abstract

We introduce a new tool, called the orbit automaton, that describes the action of an automaton group $G$ on the subtrees corresponding to the orbits of $G$ on levels of the tree. The connection between $G$ and the groups generated by the orbit automata is used to find elements of infinite order in certain automaton groups for which other methods failed to work.