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Erschienen in:
2012 | OriginalPaper | Buchkapitel
Automaton Ranks of Some Self-similar Groups
verfasst von : Adam Woryna
Erschienen in: Language and Automata Theory and Applications
Verlag: Springer Berlin Heidelberg
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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.