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 admits a left-invariant total order, and is a locally nilpotent subgroup of , then a left-invariant total order on can be chosen so that its restriction to is both left-invariant and right-invariant. Both results follow from recurrence properties of the action of on its binary relations.
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Peter Linnell, Dave Witte Morris, Amenable groups with a locally invariant order are locally indicable. Groups Geom. Dyn. 8 (2014), no. 2, pp. 467–478DOI 10.4171/GGD/234