Geometriae Dedicata

, Volume 183, Issue 1, pp 195–213

Affine automorphisms of rooted trees

Original Paper

DOI: 10.1007/s10711-016-0154-4

Cite this article as:
Savchuk, D.M. & Sidki, S.N. Geom Dedicata (2016) 183: 195. doi:10.1007/s10711-016-0154-4


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 \({\mathbb {Z}}_2^2\wr {\mathbb {Z}}\).


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

Mathematics Subject Classification

20E08 20F65 

Funding information

Funder NameGrant NumberFunding Note
Simons Foundation
  • 317198
University of South Florida
  • Proposal Enhancement Grant

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  2. 2.Departamneto de MatematicaUniversidade de BrasíliaBrasíliaBrazil

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