Left-orderable groups that don't act on the line

TL;DR

This paper constructs examples of left-orderable groups that cannot act on the real line, demonstrating limitations of group actions and embeddings in the context of homeomorphism groups.

Contribution

It provides the first example of a left-orderable group of the same size as Homeo+(R) that does not embed in it, and introduces new techniques for analyzing group actions.

Findings

The group of germs at infinity of orientation-preserving homeomorphisms of R admits no action on the line.

Constructed a finitely generated group of germs that does not extend to Homeo+(R).

Extended a theorem of E. Militon on homomorphisms between groups of homeomorphisms.

Abstract

We show that the group of germs at infinity of orientation-preserving homeomorphisms of R admits no action on the line. This gives an example of a left-orderable group of the same cardinality as Homeo+(R) that does not embed in Homeo+(R). As an application of our techniques, we construct a finitely generated group of germs that does not extend to Homeo+(R) and, separately, extend a theorem of E. Militon on homomorphisms between groups of homeomorphisms.