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Affine automorphisms of rooted trees

Abstract

We introduce a class of automorphisms of rooted d-regular trees arising from affine actions on their boundaries viewed as infinite dimensional modules \({\mathbb {Z}}_d^{\infty }\). 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 \({\mathbb {Z}}_2^2\wr {\mathbb {Z}}\).

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Acknowledgments

We would like to thank the anonymous referee for careful reading of the manuscript and for useful comments that enhanced the exposition of the paper. The first author is also grateful to Rostislav Grigorchuk for fruitful discussions that stimulated the development of this work.

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Correspondence to Dmytro M. Savchuk.

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Dmytro M. Savchuk Partially Supported by the Proposal Enhancement Grant from University of South Florida and the Simons Collaboration Grant #317198 from Simons Foundation.

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Savchuk, D.M., Sidki, S.N. Affine automorphisms of rooted trees. Geom Dedicata 183, 195–213 (2016). https://doi.org/10.1007/s10711-016-0154-4

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  • DOI: https://doi.org/10.1007/s10711-016-0154-4

Keywords

  • Automorphisms of rooted trees
  • Self-similar groups
  • Affine actions
  • Lamplighter group
  • Automata groups

Mathematics Subject Classification

  • 20E08
  • 20F65