Robert Young University of Toronto Lipschitz maps from n-balls to the (2n-1)-dimensional Heisenberg group H_n (with a sub-Riemannian metric) are abundant, but Lipschitz maps from higher-dimensional balls are rare. That is, any Lipschitz (n-1)-sphere can be filled by a Lipschitz n-ball, but most n-spheres can't be filled by (n+1)-balls. What about higher dimensions? In this talk, we'll describe the Lipschitz homotopy groups of the Heisenberg group and construct fractals in H_n that fill some higher-dimensional spheres. Joint work with Stefan Wenger.