Volume-preserving actions of simple algebraic Q-groups on low-dimensional manifolds

Abstract

We prove that SL(n,Q) has no nontrivial, C-infinity, volume-preserving action on any compact manifold of dimension strictly less than n. More generally, suppose G is a connected, isotropic, almost-simple algebraic group over Q, such that the simple factors of every localization of G have rank at least two. If there does not exist a nontrivial homomorphism from G(R) to GL(d,C), then every C-infinity, volume-preserving action of G(Q) on any compact d-dimensional manifold must factor through a finite group.