Lattices in semisimple groups and distal geometric structures (Q1079672)

A connected closed subgroup of GL(n,\({\mathbb{R}})\) is called distal if G acts distally on \({\mathbb{R}}^ n\) (i.e. \(g_ n(v)\to 0\) for some sequence \(g_ n\in G\) implies \(v=0)\). A distal struture on a vector bundle means a reduction of the structure group of the bundle to a distal linear group. Now, let \(\Gamma\) be a discrete Kazhdan group (i.e. it satisfies Kazhdan's ''property T''). The main result of the paper is the following theorem. Suppose that \(\Gamma\) acts smoothly and ergodically on a compact manifold. Then \(\Gamma\) leaves a smooth Riemannian metric invariant. Hence, there is a compact Lie group K, a closed subgroup \(K_ 0\subset K\), and a homomorphism \(\phi\) : \(\Gamma\) \(\to K\) with dense image, such that the \(\Gamma\) action is smoothly conjugate to the action of \(\Gamma\) on \(K/K_ 0\) defined by \(\phi\). As a corollary one gets the following result. Any regular distal ergodic action of a Kazhdan group on a smooth compact manifold is isometric.