Slow entropy and differentiable models for infinite-measure preserving Z ^k actions

We define slow entropy invariants for Z ^d actions on infinite measure spaces, which measure growth of itineraries at subexponential scales. We use this notion to construct infinite-measure preserving Z ² actions which cannot be realized as a group of diffeomorphisms of a compact manifold preserving a Borel measure, in contrast to the situation for Z actions, where every infinite-measure preserving action can be realized in this way.