The Trans-Carpathian Seminar on Geometry & Physics

A Carnot group G is naturally equipped with equivalent Euclidean and subriemannian metrics. Gromov has asked the following question for submanifolds of G: Determine all possible pairs (A,B) of real numbers such that there exists a submanifold M of G with Euclidean Hausdorff dimension A and sub-Riemannian Hausdorff dimension B.To answer this question in general is difficult because the structure of the underlying Lie algebra is significant. If we consider Gromov's questions for subsets of G, then a complete answer can be formulated. The solution uses elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. An interesting bi-product of this work is a relatively simple method for calculating dimensions of nonlinear iterated function systems. This is the result of joint work with Zoltan Balogh (Bern) and Jeremy Tyson (Illinois).