This article is a brief summary of the lecture delivered at the RIMS workshop, July 2005. We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished phenomenon here is that even a 'small' deformation as discrete subgroups may not preserve the condition of properly discontinuous actions. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concept 'stability' and 'local rigidity' of discontinuous groups for homogeneous spaces. As a test case, we provide a concrete and explicit description of the deformation space of Z^k acting properly discontinuously on R^k+1 by affine nilpotent transformations. This is carried out by characterizing the set of properly discontinuous groups in the deformation space of discrete subgroups.

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