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Journal Articles Journal of Algebra Year : 2020

Infinite Automaton Semigroups andGroups Have Infinite Orbits

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Daniele d'Angeli
  • Function : Author
Emanuele Rodaro
  • Function : Author
Jan Philipp Wächter
  • Function : Author

Abstract

We show that an automaton group or semigroup is infinite if and only if it admits an ω-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 ω -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.
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Dates and versions

ensl-02409431 , version 1 (11-12-2020)

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Daniele d'Angeli, Dominik Francoeur, Emanuele Rodaro, Jan Philipp Wächter. Infinite Automaton Semigroups andGroups Have Infinite Orbits. Journal of Algebra, 2020, 553, pp.119-137. ⟨10.1016/j.jalgebra.2020.02.014⟩. ⟨ensl-02409431⟩

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