Rigidity of 2-step carnot groups

Abstract

In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo Hand <i>J</i>-type algebras are given. In particular, we establish the relation of the so-called <i>J</i>²-condition to rigidity, and we explore these conditions in relation to pseudo <i>H</i>-type algebras.