Automaton Ranks of Some Self-similar Groups
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.
KeywordsTree of Words Self-similar Group Automaton Group Wreath Product
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- 8.Kharlampovich, O., Khoussainov, B., Miasnikov, A.: From automatic structures to automatic groups, 35 pages (2011), arXiv:1107.3645v2 [math.GR] Google Scholar
- 11.Nekrashevych, V., Sidki, S.: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms. In: Groups: Topological, Combinatorial and Arithmetic Aspects. LMS Lecture Notes Series, vol. 311, pp. 375–404 (2004)Google Scholar