On minimal actions of finite simple groups on homology spheres and Euclidean spaces

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2, p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.