Construct nontrivial actions disjoint from all ergodic actions for general countable groups

Construct a non-identity invertible probability measure-preserving action of an arbitrary countable group G that is disjoint from every ergodic probability measure-preserving G-action; that is, explicitly produce a nontrivial G-action whose only joinings with any ergodic G-action are the product measures.

Background

The paper gives a direct proof (extending to any countable group) of a characterization of systems that are disjoint from all ergodic systems. While nontrivial examples are easy in the classical Z-setting (e.g., the identity on a nonatomic space), constructing such examples for general groups is not straightforward.

The authors prove that for every countable infinite amenable group, there exist non-identity actions disjoint from all ergodic actions, and they also present a specific construction for F2. However, they explicitly note in the Introduction that for a general group they do not know how to construct nontrivial examples, highlighting a gap between the general characterization and explicit constructions across all groups.