Hans Martin Reimann

Rigidity of H–type groups

Abstract. On H–type groups N the left invariant horizontal vector fields span a subbundle of the tangent bundle, called the horizontal bundle HN. Generalized contact mappings f on N are smooth mappings which preserve HN. The question is: how many such mappings exist? In the case of the Heisenberg group these are the contact mappings in the classical sense and they exist in abundance. In this paper it is shown that if the dimension of the center of N is at least three, then the generalized contact mappings are in the automorphism group of a finite dimensional Lie algebra g. The elements in

Functions of Bounded mean Oscillation and Quasiconformal Mappings

{\vec g}

The complex cross ratio on the Heisenberg group

are the infinitesimal generators of local one parameter subgroups of generalized contact transformations. Rigidity is defined as the property that

Contact and Conformal Maps in

{\vec g}

Contact and conformal maps on Iwasawa N groups

is finite dimensional. For the case of the complexified Heisenberg group, i.e. the case when the dimension of the center of N is two, it has been shown [RR] that g is infinite dimensional.