Coset-preserving maps on nilpotent Lie groups

Abstract:

Rigidity theorems tell us when an apparently weak similarity of objects implies a stronger similarity of objects. If the Heisenberg group, a three-dimensional nilpotent Lie group, is mapped to itself bijectively, such that cosets are mapped to cosets, just how close is the map to being an isomorphism? After outlining the proof of this interesting result, we consider coset-preserving maps on any nilpotent Lie groups, higher dimensional versions of the Heisenberg group, and extend our rigidity theorem by induction.