I will report on recent work concerning Hausdorff dimension distortion by Sobolev and QC mappings in foliated metric spaces. We will study the question of estimating the size of the fibers of the foliation whose Hausdorff dimension of the image under a Sobolev mapping is raised above a certain threshold. Main examples include Euclidean spaces foliated by lower dimensional subspaces and Heisenberg groups foliated by homogeneous subgroups.
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