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Local rigidity for certain groups of toral automorphisms. (English) Zbl 0785.22012

In 1984, R. J. Zimmer has raised the question of infinitesimal and local rigidity for the action of \(\Gamma=SL(n,\mathbb{Z})\) on \(\mathbb{T}^ n\), \(n\geq 3\) [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 2, 1247- 1258 (1987; Zbl 0671.57028)]. A first result in this direction has been obtained in 1991 by the second author of the paper under review [Trans. Am. Math. Soc. 324, 421-445 (1991; Zbl 0726.57028)]. Specifically, he has shown that the action of \(\Gamma\) on \(\mathbb{T}^ n\) by automorphisms is infinitesimally rigid for \(n\geq 7\). In the paper under review, a stronger result regarding Zimmer’s question is proved. Specifically, it is demonstrated that the standard action of \(\Gamma\) on \(\mathbb{T}^ n\) is locally rigid for \(n\geq 4\). In the course of the proof, a global rigidity result for actions of free abelian subgroups of maximal rank in \(SL(n,\mathbb{Z})\) is also obtained.
Reviewer: D.Savin (Montreal)

MSC:

22E40 Discrete subgroups of Lie groups
57S20 Noncompact Lie groups of transformations
58A99 General theory of differentiable manifolds
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