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Affine automorphisms of rooted trees. (English) Zbl 1387.20021
Summary: We introduce a class of automorphisms of rooted d-regular trees arising from affine actions on their boundaries viewed as infinite dimensional modules \(\mathbb Z^\infty_d\). This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from this class and show that it is isomorphic to the index 2 extension of the rank 2 lamplighter group \(\mathbb Z^2_2 \wr \mathbb Z\).

20E08 Groups acting on trees
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E22 Extensions, wreath products, and other compositions of groups
20B27 Infinite automorphism groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI arXiv
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