Infinitesimal regidity for smooth actions of discrete subgroups of Lie groups (Q910773)

The author examines some rigidity properties of nonisometric actions of a cocompact discrete subgroup \(\Gamma\) of a connected noncompact semisimple Lie group G with finite center and neither compact nor simple factors locally isomorphic to any SO(1,n) or SU(1,n) on a compact Riemannian manifold M (for isometric action see \textit{R. Zimmer} [Publ. Math. Inst. Hautes Études Sci. 59, 5-33 (1984; Zbl 0576.22013)]). Namely, such ergodic \(\Gamma\)-action on M is \(L^ 2\)-infinitesimally rigid, i.e. the map \(H^ 1(\Gamma,Vect(M))\to H^ 1(\Gamma,Vect_{L^ 2}(M))\) is zero. Moreover, if additionally for a homomorphism \(\pi\) : \(G\to H\), \(M=H/\Lambda\), with semisimple H, we have either 1) \(\pi\) (\(\Gamma)\) is dense in H, or 2) \(H=H_ 1\times H_ 2\) and \(\pi\) (\(\Gamma)\) projects densely into \(H_ 1\) and trivially into \(H_ 2\), then the \(\Gamma\)- action on M is infinitesimally rigid, i.e. \(H^ 1(\Gamma,Vect(M))=0\), where Vect(M) is the space of smooth vector fields on M.