Geometric lattice actions, entropy and fundamental groups

Let Gamma be a lattice in a noncompact simple Li Group G, where R - rank(G) > 2. Suppose Gamma acts analytically and ergodically on a compact manifold M preserving a unimodular rigid geometric structure (e.g. a connection and a volume). We show that either the F action is isometric or there exists a "large image" linear representation sigma of pi(1) (M). Under an additional assumption on the dynamics of the action, we associate to sigma a virtual arithmetic quotient of full entropy.