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

Abstract

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