The invariant random order extension property is equivalent to amenability

TL;DR

This paper establishes that a countable group admits an invariant random total order extension of any invariant random partial order if and only if the group is amenable, linking order extension properties to group amenability.

Contribution

It proves the equivalence between the invariant random order extension property and amenability for countable groups, providing a new characterization of amenability.

Findings

Invariant random partial orders can be extended to total orders iff the group is amenable.

Non-amenable groups lack the invariant random order extension property.

The result generalizes previous examples and characterizations of group properties.

Abstract

Recently, Glasner, Lin and Meyerovitch gave a first example of a partial invariant order on a certain group that cannot be invariantly extended to an invariant random total order. Using their result as a starting point we prove that any invariant random partial order on a countable group could be invariantly extended to an invariant random total order iff the group is amenable.