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Deforming discontinuous subgroups for threadlike homogeneous spaces.
- Source :
- Geometriae Dedicata; Jun2010, Vol. 146 Issue 1, p117-140, 24p
- Publication Year :
- 2010
-
Abstract
- Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space $${\fancyscript{M}=G/H}$$ and any deformation of Γ, the deformed discrete subgroup may utterly destroy its proper discontinuous action on $${\fancyscript{M}}$$ as H is not compact (except the case when it is trivial). To understand this specific issue, we provide an explicit description of the parameter and the deformation spaces of any abelian discrete Γ acting properly discontinuously and fixed point freely on G/ H for an arbitrary H of a threadlike nilpotent Lie group G. The topological features of deformations, such as the local rigidity and the stability are also discussed. Whenever the Clifford-Klein form Γ\ G/ H in question is assumed to be compact, these spaces are cutely determined and unlike the case of Heisenberg groups, the deformation space fails in general to be a Hausdorff space. We show further that this space admits a smooth manifold as its open dense subset. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00465755
- Volume :
- 146
- Issue :
- 1
- Database :
- Complementary Index
- Journal :
- Geometriae Dedicata
- Publication Type :
- Academic Journal
- Accession number :
- 50423471
- Full Text :
- https://doi.org/10.1007/s10711-009-9429-3