On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces
type: Item
instance of: scholarly article
title: On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces
zbMATH Open document ID: 1244.22005
DOI: 10.1134/S000143461105018X
author: Juan-Miguel Gracia
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: Let \(G\) be an (\(n\)-step) threadlike Lie group, i.e., its Lie algebra admits a basis \(\{X,Y_1,\dots,Y_n\}\) with nontrivial Lie brackets \([X,Y_i]=Y_{i+1}\) (for \(1\leq i \leq n-1\)). Let \(G/H\) be a homogeneous space, and let \(\Gamma\) be a non-abelian discontinuous subgroup of \(G\) acting on \(G/H\). The problem of describing deformations was introduced by \textit{T. Kobayashi} [J. Geom. Phys. 12, No. 2, 133--144 (1993; Zbl 0815.57029)]. In the present paper, the authors show that any non-abelian discrete subgroup is stable. One consequence is that any resulting deformation space \({\mathcal I} (\Gamma, G, H)\) is a Hausdorff space, where \(\Gamma\) acts on the threadlike homogeneous space \(G/H\) as a discontinuous subgroup. Whenever the rank of \(\Gamma\) is bigger than three, this space is also shown to be endowed with a smooth manifold structure. But if \(k=3\), then \({\mathcal I} (\Gamma, G, H)\) admits a smooth manifold structure as its open dense subset.
reviewed by: Sergio Console
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 5980825
zbMATH Keywords: threadlike Lie group
zbMATH Keywords: stable subgroup
zbMATH Keywords: discrete subgroup
zbMATH Keywords: deformation space
zbMATH Keywords: rigidity
zbMATH Keywords: Lie algebra
zbMATH Keywords: discontinuous action
zbMATH Keywords: smooth manifold
zbMATH Keywords: Hausdorff space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5980825