Extensions of invariant random orders on groups

Abstract

In this paper, we study the action of a countable group $\Gamma$ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group, the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner–Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on ${\mathrm{SL}}_3(\mathbb{Z})$ corresponding to the semigroup generated by the standard unipotents cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended.