Actions of discrete linear groups and Zimmer's conjecture
Let a lattice group \(\Gamma\) of a higher rank semisimple Lie group \(G\) act by bundle automorphisms on a principal \(H\)-bundle \(P\) over a smooth compact manifold \(M\). Assuming that the action of \(\Gamma\) preserves a \(C^r\) connection for some \(r \geq 0\), does not preserve a measurable Riemannian metric and that the smooth measure associated with the \(\Gamma\)-invariant volume form has countably many ergodic components, it is proved that in general this action can be derived from two classical types. More precisely, one has: Either the Zariski closure of the connection's full holonomy group contains a nontrivial homomorphic image of \(G\) or the action is \(C^{r+2}\) isomorphic to: (1) the affine action on flat manifolds and tori (then \(\Gamma\) is a lattice subgroup of \(\mathbb{R}^d\) or a subgroup of \(\text{SL}(d,\mathbb{Z}))\); (2) left translations on compact quotients of non-compact semisimple Lie groups.
scientific article; zbMATH DE number 892614