Pertti Mattila University of Helsinki According to Marstrand's projection and slicing theorems Hausdorff dimension behaves as expected when a subset of a Euclidean space is projected into planes or intersected with planes. In Heisenberg groups such a precise information is not possible. I shall tell about some partial results. This is joint work with Z. Balogh, E. Durand Cartagena, K. F\"assler and J. Tyson.