On elementary amenable bounded automata groups

TL;DR

This paper investigates elementary amenable bounded automata groups, classifying them within natural subclasses and showing that certain iterated monodromy groups are either virtually abelian or not elementary amenable.

Contribution

It identifies and classifies elementary amenable bounded automata groups within specific subclasses, clarifying their structure and properties.

Findings

Iterated monodromy groups of post-critically finite polynomials are either virtually abelian or not elementary amenable.

Elementary amenable bounded automata groups can be isolated in three natural subclasses.

The study advances understanding of the structure of elementary amenable groups acting on rooted trees.

Abstract

There are several natural families of groups acting on rooted trees for which every member is known to be amenable. It is, however, unclear what the elementary amenable members of these families look like. Towards clarifying this situation, we here study elementary amenable bounded automata groups. We are able to isolate the elementary amenable bounded automata groups in three natural subclasses of bounded automata groups. In particular, we show that iterated monodromy groups of post-critically finite polynomials are either virtually abelian or not elementary amenable.