# Infinite Automaton Semigroups and Groups Have Infinite Orbits

## TL;DR

This paper proves that automaton groups and semigroups are infinite if and only if they have an infinite orbit on some right-infinite word, providing a new characterization and implications for the finiteness problem.

## Contribution

It establishes a necessary and sufficient condition for infiniteness of automaton groups and semigroups based on infinite orbits, generalizing previous results.

## Key findings

- Automaton groups and semigroups are infinite iff they have an infinite orbit on an ω-word.
- Finitely generated subgroups and subsemigroups are infinite under the same condition.
- The result connects automaton semigroups with their duals and impacts the algorithmic finiteness problem.

## Abstract

We show that an automaton group or semigroup is infinite if and only if it admits an $\omega$-word (i. e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups are infinite if and only if there is an $\omega$ -word with an infinite orbit under their action. The proof also shows some interesting connections between the automaton semigroup and its dual. Finally, our result is interesting from an algorithmic perspective as it allows for a reformulation of the finiteness problem for automaton groups and semigroups.