Finite-state self-similar actions of nilpotent groups |
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Authors: | Ievgen V Bondarenko Rostyslav V Kravchenko |
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Institution: | 1. Mechanics and Mathematics Department, National Taras Shevchenko University of Kyiv, vul. Volodymyrska 64, 01033, Kyiv, Ukraine 2. Department of Mathematics, University of Chicogo, Chicago, IL, 60637, USA
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Abstract: | 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. |
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