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A new approach to the rigidity of discrete group actions. (English) Zbl 0874.58006

This is a comprehensive study of the local structure of moduli spaces of group actions, in particular the local rigidity. The author confines himself to smooth actions of discrete groups on differentiable manifolds. In particular, he is interested in the property of \(k\)-rigidity, which requires that a perturbation of an action which is sufficiently small with respect to the \(C^k\)-topology, cannot change the action in an essential manner, or in other words, a sufficiently small neighborhood of the action shrinks to a single point when projected to the moduli space. The main result of the paper asserts that for every \(n\geq 21\), the standard action of any cocompact lattice of \(\text{SL}_{n+1}\mathbb{R}\) on the \(n\)-sphere is 4-rigid.

MSC:

58D19 Group actions and symmetry properties
58E40 Variational aspects of group actions in infinite-dimensional spaces
22E40 Discrete subgroups of Lie groups
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References:

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