Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher-rank lattices
Let \(\pi\) be the standard action of \(SL(n,\mathbb{Z})\) on the torus \(\mathbb{T}^n\), \(n \geq 3\). For a positive integer \(d\), consider the action \(\rho_0\) of \(SL(n,\mathbb{Z})\) on \(\mathbb{T}^{n + d} = \mathbb{T}^n \times \mathbb{T}^d \) given by \(\rho_0 (A) (x,y) = (\pi (A) x,y)\) (where \(x \in \mathbb{T}^n\), \(y \in \mathbb{T}^d\) and \(A \in SL(n,\mathbb{Z}))\). Then \(\rho_0\) is a nonirreducible and nontransitive partially hyperbolic action. The authors prove that, for any \(\infty > K \geq 4\), the action \(\rho_0\) is \(C^{K + Lip,K - 3}\)-deformation rigid. \textit{A. N. Livšic} [Math. USSR, Izv. 6(1972), 1278-1301 (1974; Zbl 0273.58013)] proved results relating closing conditions for topologically transitive \(C^1\) Anosov diffeomorphisms on a compact Riemannian manifold to cocycles on the manifold with values in a finite-dimensional nilpotent Lie group. The authors prove an extension of Livšic's theorem to cocycles with values in the group of \(C^K\)-diffeomorphisms of a compact closed manifold with a stably trivial tangent bundle (here as above \(4 \leq K < \infty)\). The application of this Livšic-type result to the deformation rigidity result requires an extension to \(SL(n,\mathbb{Z})\)-actions instead of \(\mathbb{Z}\)-actions. The authors also prove and use an extension of the Hirsch-Pugh-Shub structural stability theorem.
scientific article; zbMATH DE number 878414
scientific article