Low dimensional actions of semisimple groups

For a simple non-compact Lie groupG with finite center we determine the smallest integern(G) such thatG has an almost effective action on a compact manifold of dimensionn(G) and characterize the compact manifolds of dimensionn(G) on whichG acts. We study actions of a semisimple groupG on compact manifolds of dimensionn(G)+1 and determine the orbit structure of the action ofG and its maximal compact subgroup. We give several examples to illustrate the results. This work was supported by an NSF postdoctoral Research Fellowship. An erratum to this article is available at .