Variants of stability of discontinuous groups for Euclidean motion groups. (English) Zbl 1370.57016
Let \(G\) be the Euclidean motion group \(O_n(\mathbb R) \cdot \mathbb R^n\) and \(H\) be a closed Lie subgroup in \(G\). The natural actions of discrete subgroups \(\Gamma \subset G\) on the homogeneous spaces \(G/H\) are considered. The problem of stability of these actions is investigated. Three variants of the notion of stability are considered: stability, introduced by T. Kobayashi and S. Nasrin [Int. J. Math. 17, No. 10, 1175–1193 (2006; Zbl 1124.57015)], geometrical stability and nearly stability.
Let \(\mathcal R(\Gamma,G,H)\) be the space of all \(\phi \in \mathrm{Hom}(\Gamma,G)\) such that the homomorphism \(\phi\) is injective, \(\phi(\Gamma)\) is discrete and acts on \(G/H\) properly and without fixed points. Let \(\mathrm{Hom}^0_d(\Gamma,G)\) be the space of homomorphisms \(\phi \in \mathrm{Hom}( \Gamma,G)\), which are injective and \(\phi(\Gamma)\) is discrete. The main result of this article is as follows. Any point of \(\mathcal R(\Gamma,G,H)\) is nearly stable, i.e. \(\mathcal R(\Gamma,G,H)\) is an open set in \(\mathrm{Hom}^0_d(\Gamma,G)\). For crystallographic discrete subgroups \(\Gamma\) it is proved that they are stable, i.e. \(\mathcal R(\Gamma,G,H)=\mathrm{Hom}^0_d(\Gamma,G)\) is open in \(\mathrm{Hom}(\Gamma,G)\).
Also a table which summarizes the obtained results for different variants of the notion of stability in the case of the Euclidean motion group is presented.
Let \(\mathcal R(\Gamma,G,H)\) be the space of all \(\phi \in \mathrm{Hom}(\Gamma,G)\) such that the homomorphism \(\phi\) is injective, \(\phi(\Gamma)\) is discrete and acts on \(G/H\) properly and without fixed points. Let \(\mathrm{Hom}^0_d(\Gamma,G)\) be the space of homomorphisms \(\phi \in \mathrm{Hom}( \Gamma,G)\), which are injective and \(\phi(\Gamma)\) is discrete. The main result of this article is as follows. Any point of \(\mathcal R(\Gamma,G,H)\) is nearly stable, i.e. \(\mathcal R(\Gamma,G,H)\) is an open set in \(\mathrm{Hom}^0_d(\Gamma,G)\). For crystallographic discrete subgroups \(\Gamma\) it is proved that they are stable, i.e. \(\mathcal R(\Gamma,G,H)=\mathrm{Hom}^0_d(\Gamma,G)\) is open in \(\mathrm{Hom}(\Gamma,G)\).
Also a table which summarizes the obtained results for different variants of the notion of stability in the case of the Euclidean motion group is presented.
Reviewer: V. V. Gorbatsevich (Moskva)
MSC:
| 57S30 | Discontinuous groups of transformations |
Citations:
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