Amenable groups with a locally invariant order are locally indicable

TL;DR

This paper proves that amenable groups with a locally invariant order are necessarily locally indicable and explores how total orders can be extended to subgroups with specific invariance properties, using recurrence properties of group actions.

Contribution

It establishes that amenable groups with locally invariant orders are locally indicable and shows how to extend total orders to certain subgroups with invariance properties.

Findings

Amenable groups with locally invariant orders are locally indicable.

Total orders on groups can be extended to subgroups with invariance properties.

Recurrence properties of group actions underpin these results.

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.