# Orbit Expandability of Automaton Semigroups and Groups

## TL;DR

This paper introduces the concept of expandability in automaton semigroups and groups, providing algorithms to decide expandability, characterizations for groups, and bounds on orbit-increasing suffixes, advancing understanding of automaton actions.

## Contribution

It defines expandability for automaton semigroups and groups, develops decision algorithms, and offers algebraic characterizations, especially for invertible automata and automaton groups.

## Key findings

- Decidability of k-expandability in exponential nondeterministic space.
- Characterization of expandability in automaton groups via shifted stabilizer.
- Every word is expandable in reversible and complete automata.

## Abstract

We introduce the notion of expandability in the context of automaton semigroups and groups: a word is k-expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k. This definition is motivated by the question which {\omega}-words admit infinite orbits: for such a word, every prefix is expandable. In this paper, we show that, on input of a word u, an automaton T and a number k, it is decidable to check whether u is k-expandable with respect to the action of T. In fact, this can be done in exponential nondeterministic space. From this nondeterministic algorithm, we obtain a bound on the length of a potential orbit-increasing suffix x. Moreover, we investigate the situation if the automaton is invertible and generates a group. In this case, we give an algebraic characterization for the expandability of a word based on its shifted stabilizer. We also give a more efficient algorithm to decide expandability of a word in the case of automaton groups, which allows us to improve the upper bound on the maximal orbit-increasing suffix length. Then, we investigate the situation for reversible (and complete) automata and obtain that every word is expandable with respect to these automata. Finally, we give a lower bound example for the length of an orbit-increasing suffix.