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Deformation of properly discontinuous actions of \(\mathbb Z^{k}\) on \(\mathbb R^{k+1}\). (English) Zbl 1124.57015
Authors’ summary: 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 \(\mathbb Z^{k}\) acting properly discontinuously on \(\mathbb R^{k+1}\) by affine nilpotent transformations. Our method uses an idea of ’continuous analogue’ and relies on the criterion of proper actions on nilmanifolds.

MSC:
57S30 Discontinuous groups of transformations
22E25 Nilpotent and solvable Lie groups
22E40 Discrete subgroups of Lie groups
53C30 Differential geometry of homogeneous manifolds
58H15 Deformations of general structures on manifolds
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