The effect of projections on dimension in the Heisenberg group

Jeremy Tyson

(University of Illinois)

Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 10, 2011, time: 10:40

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. The semidirect product decomposition of the Heisenberg group into a horizontal subgroup and a complementary vertical subgroup induces a pair of projection-type mappings. We study the effect of these mappings on the sub-Riemannian Hausdorff dimensions of sets. We establish almost sure dimension estimates phrased in terms of the isotropic Grassmannian of $m$-dimensional horizontal subgroups of the Heisenberg group. Our results generalize classical almost sure dimension theorems of Marstrand, Kaufman and Mattila from the Euclidean setting. This talk is based on joint work with Zoltán Balogh, Estibalitz Durand Cartagena, Katrin Fässler and Pertti Mattila.