Classification of some cohomologically $C^0$-stable continuous group actions on metric spaces

After Katok, a homeomorphism $f\colon M\to M$ of a compact metric space is said to be cohomologically $C^0$-stable if its space of real $C^0$-coboundaries is closed in $C^0(M)$. Kocsard proved that this is the case if and only if $f$ is periodic. We extend the classification to actions of arbitrary finitely generated groups: an action $\alpha : G \to Homeo(M)$ is cohomologically $C^0$ - stable if and only if the image $\alpha(G)$ is a finite group. In particular this settles the case of $\mathbb Z^k$-actions generated by finitely many commuting homeomorphisms. Notably, no amenability assumption is needed: we explain why spectral-gap phenomena for non-amenable actions, which do produce cohomological stability in H\"older, Sobolev and $L^2$ categories, are invisible to the uniform norm. We also discuss the genuinely different smooth category and state a conjecture regarding cohomological $C^\infty$-stability of $\mathbb Z^k$-action by smooth circle diffeomorphisms without periodic orbits, connecting the problem with works of Moser, Fayad-Khanin, Avila-Kocsard and Petkovi\'c.