Kobayashi, Toshiyuki; Nasrin, Salma Deformation of properly discontinuous actions of \(\mathbb Z^{k}\) on \(\mathbb R^{k+1}\). (English) Zbl 1124.57015 Int. J. Math. 17, No. 10, 1175-1193 (2006). 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. Reviewer: Emil Molnár (Budapest) Cited in 3 ReviewsCited in 21 Documents 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 Keywords:discontinuous group; deformation; rigidity; proper action; affine transformation; properly discontinuous action; homogeneous space PDFBibTeX XMLCite \textit{T. Kobayashi} and \textit{S. Nasrin}, Int. J. Math. 17, No. 10, 1175--1193 (2006; Zbl 1124.57015) Full Text: DOI arXiv 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.