The invariant random order extension property is equivalent to amenability

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.

27 pages. Some corrections and expanded discussion following the referee report