Rigidity of isometric lattice actions on compact Riemannian manifolds (Q1584717)

Let \(G\) be a simple Lie group with finite centre of real rank \(\geq 2\) and let \(\Gamma\) be a cocompact lattice in \(G\). The main result of the paper is the local rigidity of any isometric action \(\rho\) of \(\Gamma\) on a compact Riemannian manifold \(Z\). More precisely, if \(\{\gamma_1,\ldots,\gamma_k\}\) is a finite generating set for \(\Gamma\), then any action \(\rho'\) of \(\Gamma\) by diffeomorphisms of \(Z\) such that all \(\rho'(\gamma_i)\) are sufficiently close to \(\rho(\gamma_i)\) in the \(C^{\infty}\)-topology is conjugate to \(\rho\) in \(\text{Diff}Z\). The proof uses the Spencer-Guillemin-Sternberg theory of deformation of structures applied to certain foliated manifolds associated to actions.