Amenable groups with a locally invariant order are locally indicable

Abstract

We show that every amenable group with a locally invariant partial order has a left-invariant total order (and is therefore locally indicable). We also show that if a group $G$ admits a left-invariant total order, and $H$ is a locally nilpotent subgroup of $G$, then a left-invariant total order on $G$ can be chosen so that its restriction to $H$ is both left-invariant and right-invariant. Both results follow from recurrence properties of the action of $G$ on its binary relations.