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

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.

19 pages, 9 figures; example with Bellaterra group added; references updated