Rigidity of higher rank Abelian cocycles with values in diffeomorphism groups (Q2642678)

In this paper, the authors prove new rigidity properties for cocycles over certain hyperbolic \(\mathbb{R}^k\) actions, \(k\geq 2\) with values in a Lie group or a diffeomorphism group sufficiently smooth and close to the identity on a set of generators. More precisely, they consider Cartan actions of SL\((n,\mathbb{R})/\Gamma\) or SL\((n,\mathbb{C})/\Gamma\), for \(n\geq 3\) and \(\Gamma\) a torsion free cocompact lattice. The paper is part of a program whose aim is to classify Hölder and smooth cocycles over higher rank Abelian hyperbolic actions of \(\mathbb{Z}^k\) or \(\mathbb{R}^k\), \(k\geq 2\), up to cohomology. The results obtained so far show that the cocycles are rigid in the sense that they are either cohomologous to constant cocycles or to some cocycles which can be easily described and which belong to a finite set. The techniques used in this paper involve notions of \(K\)-theory, and they originate in the paper [\textit{D. Damjanovic} and \textit{A. Katok}, Discrete Contin. Dyn. Syst. 13, No. 4, 985--1005 (2005; Zbl 1109.37029)].