Extensions of invariant random orders on groups

TL;DR

This paper explores the space of invariant random orders on countable groups, demonstrating its richness, characterizing realizable actions, and providing the first example of a partial order that cannot be randomly extended.

Contribution

It introduces the concept of invariant random orders, characterizes the actions they can realize, and presents the first example of a partial order that cannot be extended to a random total order.

Findings

The space of invariant random orders contains copies of all free ergodic actions.

A Glasner-Weiss dichotomy for the simplex of invariant random orders.

The invariant partial order on SL_3(Z) cannot be extended to an invariant random total order.

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 $\mathrm{SL}_3(\mathbf{Z})$ corresponding to the semigroup of positive matrices 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.