Abstract

Given a group G and a positive integer d ≥ 2 we introduce the notion of an automaton rank of a group G with respect to its self-similar actions on a d-ary tree of words as the minimal number of states in an automaton over a d-letter alphabet which generates this group (topologically if G is closed). We construct minimal automata generating free abelian groups of finite ranks, which completely determines automaton ranks of free abelian groups. We also provide naturally defined 3-state automaton realizations for profinite groups which are infinite wreath powers … ≀ H ≀ H for some 2-generated finite perfect groups H. This determines the topological rank and improves the estimation for the automaton rank of these wreath powers. We show that we may take H as alternating groups and projective special linear groups.

Keywords

Tree of Words Self-similar Group Automaton Group Wreath Product 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adam Woryna
    • 1
  1. 1.Institute of MathematicsSilesian University of TechnologyGliwicePoland

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