Pansu pullback and rigidity of mappings between Carnot groups

We consider mappings $f:G\supset U\rightarrow G'$ where $G$ and $G'$ are Carnot groups and U is an open subset. We prove a number of new structural results for Sobolev (in particular quasisymmetric) mappings, establishing (partial) rigidity or (partial) regularity theorems, depending on the context. In particular, we prove the quasisymmetric rigidity conjecture for Carnot groups which are not rigid in the sense of Ottazzi-Warhurst.