A compact manifold with infinite-dimensional co-invariant cohomology

Let $M$ be a smooth manifold. When $\mathrm{\Gamma }$ is a group acting on the manifold $M$ by diffeomorphisms one can define the $\mathrm{\Gamma }$-co-invariant cohomology of $M$ to be the cohomology of the differential complex ${\mathrm{\Omega }}_c(M{)}_{\mathrm{\Gamma }}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\omega -{\gamma }^{\ast }\omega ,\omega \in {\mathrm{\Omega }}_c(M),\gamma \in \mathrm{\Gamma }\}\mathrm{.}$ For a Lie algebra $\mathcal{G}$ acting on the manifold $M$, one defines the cohomology of $\mathcal{G}$-divergence forms to be the cohomology of the complex ${\mathcal{C}}_{\mathcal{G}}(M)=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{L_X\omega ,\omega \in {\mathrm{\Omega }}_c(M),X\in \mathcal{G}\}\mathrm{.}$ In this short paper we present a situation where these two cohomologies are infinite dimensional.