Quasiconformal maps on model Filiform groups

We describe all quasiconformal maps on the higher (real and complex) model Filiform groups equipped with the Carnot metric, including non-smooth ones. These maps have very special forms. In particular, they are all biLipschitz and preserve multiple foliations. The results in this paper have implications to the large scale geometry of nilpotent Lie groups and negatively curved solvable Lie groups.