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The lamplighter group $$\mathbb Z_3 \wr \mathbb Z$$ generated by a bireversible automaton. (English) Zbl 1368.20033
Summary: We construct a bireversible self-dual automaton with three states over an alphabet with three letters which generates the lamplighter group $$\mathbb Z_3 \wr \mathbb Z$$. In particular, this fact shows that not all groups defined by birevirsible automata are finitely presented.

##### MSC:
 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E08 Groups acting on trees 20E22 Extensions, wreath products, and other compositions of groups 20F05 Generators, relations, and presentations of groups 68Q70 Algebraic theory of languages and automata
##### Keywords:
automaton group; bireversible automaton; lamplighter group
AutomGrp
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##### References:
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