Coarse homology of leaves

We investigate the coarse homology of leaves in foliations of compact manifolds. This is motivated by the observation that the non-leaves constructed by Schweitzer and by Zeghib all have non-finitely generated coarse homology. This led us to ask whether the coarse homology of leaves in a compact manifold always has to be finitely generated. We show that this is not the case by proving that there exist many leaves with non-finitely generated coarse homology. Moreover, we improve Schweitzer's non-leaf construction and produce non-leaves with trivial coarse homology.