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.
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