Local rigidity for certain groups of toral automorphisms (Q1196312)

In 1984, \textit{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.