Deformation of Properly Discontinuous Actions of Z^k on R^{k+1}
  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 Z^k acting properly discontinuously on R^{k+1} by affine nilpotent transformations.   Our method uses an idea of `continuous analogue' and relies on the criterion of proper actions on nilmanifolds. 
Toshiyuki Kobayashi and Salma Nasrin