Non-elementary amenable subgroups of automata groups

TL;DR

This paper establishes conditions under which certain automorphism groups of trees are not elementary amenable, providing simpler proofs for known examples and confirming a conjecture about branch groups.

Contribution

It introduces new criteria for non-elementary amenability in automorphism groups of trees and proves all finitely generated branch groups are non-elementary amenable.

Findings

All known non-elementary amenable groups acting on trees are covered.

Finitely generated branch groups are proven to be non-elementary amenable.

Provides simplified proofs for groups of intermediate growth and Basilica group.

Abstract

We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.