Orbits of Abelian Automaton Groups
Automaton groups are a class of self-similar groups generated by invertible finite-state transducers . Extending the results of Nekrashevych and Sidki , 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 .
KeywordsAutomaton 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.
- 2.Becker, T.: Embeddings and orbits of abelian automaton groups (2018). https://github.com/tim-becker/thesis-code
- 3.Bondarenko, I., et al.: Classification of groups generated by 3-state automata over a 2-letter alphabet. Algebra Discrete Math. 1, April 2008Google Scholar
- 13.Okano, T.: Invertible binary transducers and automorphisms of the binary tree. Master’s thesis, Carnegie Mellon University (2015)Google Scholar
- 16.Stein, W.: Algebraic Number Theory, a Computational Approach. Harvard, Massachusetts (2012)Google Scholar