Anti-classification results for groups acting freely on the line

TL;DR

This paper investigates the complexity of classifying countable Archimedean groups and related structures using descriptive set theory, revealing inherent limitations in their classification due to topological and set-theoretic constraints.

Contribution

It introduces the space of Archimedean left-orderings, analyzes the complexity of their isomorphism and bi-embeddability relations, and demonstrates the non-classifiability by countable sets of reals.

Findings

The isomorphism relation for countable Archimedean groups is highly complex.

The potential class of this relation is not $oldsymbol{ ext{Pi}}^0_3$, indicating non-classifiability.

Similar complexity results hold for circularly ordered groups and o-minimal structures.

Abstract

We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation induced by the natural action of $\mathrm{GL}_2(\mathbb{Q})$ on $\mathrm{Ar}(\mathbb{Q}^2)$ is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau. In particular, we show that its potential class is not $\boldsymbol{\Pi}^0_3$. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.