Groups with classifiable actions on the line
Joaqu\'in Brum, Mart\'in Gilabert Vio, Nicol\'as Matte Bon

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 of countable groups such that the conjugacy relation between minimal actions of on by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of is known. We show a number of stability properties of under group-theoretic operations and that contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group that is not in , such that is amenable if and only if Thompson's group is amenable. We also prove that the semiconjugacy relation among cocompact actions of a countable group is smooth if and only if , and that it is essentially countable even when is not finitely generated. In the Appendix, we…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
