Deformation of Properly Discontinuous Actions of Z^k on R^{k+1}

Details Deformation of Properly Discontinuous Actions of Z^k on R^{k+1} Deformation of Properly Discontinuous Actions of Z^k on R^{k+1}

2006-03-14

arxiv.org/abs/math/0603318v1

International Journal of Mathematics 17 (2006), 1175-1193

Mathematics

Differential Geometry

Scientific paper

10.1142/S0129167X06003862

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.

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