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Tree-wreathing applied to generation of groups by finite automata. (English) Zbl 1108.20025

Summary: We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable Abelian residually finite 2-group \(H\) and \(B=\mathbf B(n,\mathbb{Z})\), a canonical subgroup of finite index in \(\text{GL}(n,\mathbb{Z})\), then the restricted wreath product \(H\text{\,wr\,}B\) can be generated by finite synchronous automata on \(0,1\). This is obtained by producing a representation of \(B\) as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of \(0\)’s is trivial. The uni-triangular group \(U=\mathbf U(n,\mathbb{Z})\) is a subgroup of \(\mathbf B(n,\mathbb{Z})\) and so, \(H\text{\,wr\,}U\) also can be generated by finite synchronous automata on \(0,1\).

MSC:

20E08 Groups acting on trees
20E22 Extensions, wreath products, and other compositions of groups
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
68Q45 Formal languages and automata
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References:

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[3] DOI: 10.1016/S0021-8693(02)00050-9 · Zbl 1027.20018 · doi:10.1016/S0021-8693(02)00050-9
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