Gromov’s dimension comparison problem in Carnot groups

Abstract: Inspired by a question in Gromov’s book on the intrinsic geometry of Carnot-Caratheodory spaces (“CC spaces seen from within”), we seek sharp comparison statements relating metric notions of dimension defined by a sub-Riemannian metric and a compatible Riemannian metric. Our main result provides such comparison theorems for Carnot-Caratheodory vs. Euclidean Hausdorff and box counting dimensions on general Carnot groups. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Caratheodory balls which generalize and extend the well-known Box-Ball theorem, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer’s work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for a variety of invariant sets of nonlinear Euclidean iterated function systems of polynomial type.