Chain conditions, elementary amenable groups, and descriptive set theory

TL;DR

This paper links chain conditions in the space of marked groups with descriptive set theory, showing certain classes are co-analytic and non-Borel, and provides a new proof of the existence of non-elementary amenable groups.

Contribution

It introduces descriptive set-theoretic characterizations of chain conditions and elementary amenability in marked groups, revealing their complexity and non-Borel nature.

Findings

Chain conditions characterized by well-founded trees

Elementary amenability is equivalent to a chain condition

Existence of finitely generated amenable groups that are not elementary amenable

Abstract

We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees, and therefore, elementary amenability is equivalent to a chain condition. Our characterization again implies the set of elementary amenable groups is co-analytic and non-Borel. As corollary, we obtain a new, non-constructive, proof of the existence of finitely generated amenable groups that are not elementary amenable.