Local rigidity of certain partially hyperbolic actions of product type (Q2748988)

This paper is part of the rigidity program initiated by \textit{R. J. Zimmer} [Proc. Int. Congr. Math., Berkeley/CA, Vol. 2, 1247-1258 (1986; Zbl 0671.57028)]. The objective of Zimmer's program is to classify the smooth actions of higher-rank semisimple Lie groups and of their (irreducible) lattices on compact manifolds which can viewed as the study of homomorphisms into the infinite-dimensional group of diffeomorphisms of the compact manifold in question. NEWLINENEWLINENEWLINEIn the paper under review the authors obtain several rigidity properties of some special higher-rank Abelian product actions. The essential property on which the results are based is that of a (TNS) action introduced by \textit{A. Katok} and the authors [Ergodic Theory Dyn. Syst. 20, No. 1, 259-288 (2000; Zbl 0977.57042)]. As an example of the results proved we have the following: Let \(N\) be a compact manifold and \(\alpha : {\mathbb Z}^{k}\rightarrow \text{Diff}^{r+1}({\mathbf T}^{m})\) be a (TNS) action, where \(k\geq 2\) and \(r\geq 0\). Consider \(\rho : {\mathbb Z}^{k}\rightarrow \text{Diff}^{r+1}({\mathbf T}^{m}\times N)\) given by \(\rho(a):\equiv \alpha (a)\times \text{Id}_{N}\). Then any \(C^{r+1}\)-action \(\widehat{\rho}\) that is \(C^{1}\)-close to \(\rho\) has an invariant horizontal lamination \(\widehat{H}\) which has \(C^{(r+1)^{-}}\) leaves and \(C^{r}\)-holonomy between the center leaves of \(\widehat{\rho}\). The closeness depends on \((r+1)\)-normal hyperbolicity and \(r\)th-order center-bunching. As \(\widehat{\rho}\) converges to \(\rho\) in \(C^{r+1}\), \(\widehat{H}\) converges to \(H\) in \(C^{(r+1)^{-}}\) and the holonomy converges uniformly in \(C^{r}\) to the identity on each center leaf. NEWLINENEWLINENEWLINEThe authors also show several results on product actions of property (\(T\)) groups (see for definition and properties of Kazhdan's property (\(T\)) [see \textit{P. de la Harpe} and \textit{A. Valette}, Astérisque 175 (1989; Zbl 0759.22001)] which permits them (together with the previous result on (TNS)) to prove among other things that the action \(\alpha \times \text{Id}_{\mathbf T}\) is \(C^{2,r^{-}}\)-locally rigid (\(r\geq 1\)) whenever \(\Gamma\) is an irreducible lattice in a linear semisimple Lie group \(G\) all of whose factors have real rank at least two and \(\alpha : \Gamma \rightarrow \text{Diff}^{\infty}({\mathbf T}^{m})\) is a linear action that contains a (TNS) subaction.