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 $${\mathbb {Z}}_2^2\wr {\mathbb {Z}}$$ .