Orbit automata as a new tool to attack the order problem in automaton   groups
  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. 
Ines Klimann and Matthieu Picantin and Dmytro Savchuk
math.GR cs.FL