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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.

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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