Let be a finitely generated, amenable group. Using an idea of É Ghys, we prove that if has a nontrivial, orientation-preserving action on the real line, then has an infinite, cyclic quotient. (The converse is obvious.) This implies that if has a faithful action on the circle, then some finite-index subgroup of has the property that all of its nontrivial, finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P Linnell.
"Amenable groups that act on the line." Algebr. Geom. Topol. 6 (5) 2509 - 2518, 2006. https://doi.org/10.2140/agt.2006.6.2509