Existence of ergodic cocycles for outer actions on type III factors

Determine whether every outer action of a discrete group on a type III von Neumann factor admits a 1‑cocycle such that the cocycle‑perturbed action is ergodic, i.e., has trivial fixed‑point algebra.

Background

The paper investigates cocycle perturbations of group actions on type III factors, extending results known for type II1 factors. In the type II1 setting, Marrakchi and Vaes proved that for every outer action of a countable amenable group, one can find a 1‑cocycle making the perturbed action ergodic, with ergodic cocycles forming a dense Gδ subset of the cocycle space.

For type III factors, despite many concrete examples of ergodic actions, the general existence of an ergodic cocycle for an arbitrary outer action had remained unresolved. The present work gives an affirmative result in the state‑preserving type III1 setting under the trivial bicentralizer assumption, partially addressing this broader question.