Extensions of invariant random orders on groups (Q6619333)

Let \(\Gamma\) be a group. The space of left-invariant orders on is a zero-dimensional compact Hausdorff topological space on which \(\Gamma\) acts by conjugation. The study of this action from the point of view of topological dynamics proved to be a powerful tool, especially in the case where this action has no fixed points, that is, the group is not bi-orderable.\N\NThe central objects of the paper under review are invariant random orders, namely probability measures on the space of orders whose distribution is invariant with respect to multiplication. The authors 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. They prove a Glasner-Weiss dichotomy regarding the simplex of invariant random orders (see [\textit{E. Glasner} and \textit{B. Weiss}, Geom. Funct. Anal. 7, No. 5, 917--935 (1997; Zbl 0899.22006)]). The authors 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. Finally, they provide the first example for a partial order (deterministic or random) that cannot be randomly extended.\N\NAt the end of the introduction, the authors add the following remark. ``Subsequently to the first arXiv version of this paper, building upon our results, \textit{A. Alpeev} [``The invariant random order extension property is equivalent to amenability'', Preprint, \url{arXiv:2206.14177}] proved that any non-amenable countable group admits a partial invariant order that cannot be extended to an invariant random total order.''