Deformation of Properly Discontinuous Actions of Z on R

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a ‘small’ deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on ‘stability’ and ‘local rigidity’ of discontinuous groups. As a test case, we give an explicit description of the deformation space of Zk acting properly discontinuously on Rk+1 by affine nilpotent transformations. Our method uses an idea of ‘continuous analogue’ and relies on the criterion of proper actions on nilmanifolds.