Groups with classifiable actions on the line

TL;DR

This paper investigates the class of countable groups with smooth conjugacy relations on the line, exploring their properties, examples, and connections to amenability and harmonic actions.

Contribution

It characterizes the class of groups with smooth conjugacy relations, shows stability under operations, and provides examples and counterexamples related to amenability.

Findings

All finitely generated groups of piecewise affine homeomorphisms are in the class.

A finitely generated group not in the class is constructed, linking amenability to Thompson's group F.

The semiconjugacy relation is smooth iff the group is in the class, and is countable even when not finitely generated.

Abstract

We motivate and study the class $\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of $\mathcal{C}$ is known. We show a number of stability properties of $\mathcal{C}$ under group-theoretic operations and that $\mathcal{C}$ contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group $G$ that is not in $\mathcal{C}$, such that $G$ is amenable if and only if Thompson's group $F$ is amenable. We also prove that the semiconjugacy relation among cocompact actions of a countable group $G$ is smooth if and only if $G \in \mathcal{C}$, and that it is essentially countable even when $G$ is not finitely generated. In the Appendix, we show that there is no good analogue of the space of harmonic actions for a countable non-finitely generated group.