Extensions of automorphisms of self-similar groups

TL;DR

This paper investigates when automorphisms of self-similar groups can be extended to the entire automorphism group of an infinite rooted tree, providing conditions for continuity and examples of non-extendable cases, focusing on lamplighter groups.

Contribution

It establishes criteria for extending automorphisms of self-similar groups to the full automorphism group of the tree and analyzes specific cases like lamplighter groups.

Findings

Conditions for the existence of automorphism extensions

Examples where extensions do not exist

Continuity criteria with respect to the depth metric

Abstract

In this work we study automorphisms of synchronous self-similar groups, the existence of extensions to automorphisms of the full group of automorphisms of the infinite rooted tree on which these groups act on. When they do exist, we obtain conditions for the continuity of such extensions with respect to the depth metric, but we also construct examples of groups where such extensions do not exist. We study the case of the lamplighter group $\mathcal{L}_k = \mathbb{Z}_k \wr \mathbb{Z}$