A compact manifold with infinite-dimensional co-invariant cohomology

Abstract: Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on the manifold $M$ by diffeomorphisms one can define the $\Gamma$-co-invariant cohomology of $M$ to be the cohomology of the differential complex $\Omega_c(M)\Gamma=\mathrm{span}{\omega-\gamma^*\omega,\;\omega\in\Omega_c(M),\;\gamma\in\Gamma}.$ 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 $\mathcal{C}{\mathcal{G}}(M)=\mathrm{span}{L_X\omega,\;\omega\in\Omega_c(M),\;X\in\mathcal{G}}.$ In this short paper we present a situation where these two cohomologies are infinite dimensional.