## 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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### References:

 [1] Abels H., J. Differential Geom. 60 pp 315– [2] DOI: 10.1142/S0129167X0500317X · Zbl 1087.22007 [3] DOI: 10.2307/2118594 · Zbl 0868.22013 [4] Goldman W., J. Differential Geom. 21 pp 301– · Zbl 0591.53051 [5] DOI: 10.1007/BF01443517 · Zbl 0662.22008 [6] DOI: 10.1016/0393-0440(93)90011-3 · Zbl 0815.57029 [7] Kobayashi T., J. Lie Theory 6 pp 147– [8] DOI: 10.1007/s002080050153 · Zbl 0891.22014 [9] DOI: 10.1007/978-3-642-56478-9_8 [10] Kobayashi T., Sugaku 57 pp 267– [11] Lipsman R., J. Lie Theory 5 pp 25– [12] DOI: 10.3836/tjm/1255958192 · Zbl 1010.22011 [13] DOI: 10.2307/1970335 · Zbl 0103.01802 [14] DOI: 10.1016/S0764-4442(99)80384-X · Zbl 0896.53043 [15] DOI: 10.2307/1970495 · Zbl 0192.12802
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