Geometriae Dedicata

, Volume 163, Issue 1, pp 339–348 | Cite as

Finite-state self-similar actions of nilpotent groups

  • Ievgen V. Bondarenko
  • Rostyslav V. KravchenkoEmail author
Original Paper


Let G be a finitely generated torsion-free nilpotent group and \({\phi:H\rightarrow G}\) be a surjective homomorphism from a subgroup H < G of finite index with trivial \({\phi}\) -core. For every choice of coset representatives of H in G there is a faithful self-similar action of the group G associated with \({(G, \phi)}\). We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for \({(G, \phi)}\). These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism \(\widehat{\phi}\) of the Lie algebra of the Mal’cev completion of G.


Self-similar action Nilpotent group Finite-state action Automaton group 

Mathematics Subject Classification (2000)

20F65 20F18 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Ievgen V. Bondarenko
    • 1
  • Rostyslav V. Kravchenko
    • 2
    Email author
  1. 1.Mechanics and Mathematics DepartmentNational Taras Shevchenko University of KyivKyivUkraine
  2. 2.Department of MathematicsUniversity of ChicogoChicagoUSA

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