Orbits of Abelian Automaton Groups

  • Tim BeckerEmail author
  • Klaus Sutner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)


Automaton groups are a class of self-similar groups generated by invertible finite-state transducers [11]. Extending the results of Nekrashevych and Sidki [12], we describe a useful embedding of abelian automaton groups into a corresponding algebraic number field, and give a polynomial time algorithm to compute this embedding. We apply this technique to study iteration of transductions in abelian automaton groups. Specifically, properties of this number field lead to a polynomial-time algorithm for deciding when the orbits of a transduction are a rational relation. These algorithms were implemented in the SageMath computer algebra system and are available online [2].


Automaton groups Embedding Number field Orbits Rational relation Classification 



The authors would like to thank Eric Bach for his helpful feedback on a draft of this paper. We also thank Evan Bergeron and Chris Grossack for many helpful conversations on the results presented.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer SciencesUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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