Affine Automorphisms of Rooted Trees

TL;DR

This paper introduces a new class of automorphisms of rooted regular trees based on affine actions, connecting them to lamplighter groups and analyzing their algebraic properties, especially for binary trees.

Contribution

It defines affine automorphisms of rooted trees, characterizes their relation to automorphism normalizers, and explores their connection to lamplighter groups and automaton groups.

Findings

Affine automorphisms include many self-similar lamplighter group realizations.

For binary trees, this class matches the normalizer of spherically homogeneous automorphisms.

An automaton group example contains an index two subgroup from this class, isomorphic to an extension of a lamplighter group.

Abstract

We introduce a class of automorphisms of rooted $d$-regular trees arising from affine actions on their boundaries viewed as infinite dimensional vector spaces. 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 $\mathbb Z_2^2\wr\mathbb Z$.