Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (Q960577)

Carnot groups are simply connected nilpotent Lie groups with graded Lie algebra equipped with a left invariant metric of sub-Riemannian type. In the paper under review Gromov's dimension comparison problem is solved for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating the optimal mutual covering of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. It is shown, that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type are obtained. The main results of the paper are illustrated by jet spaces Carnot groups.