Geometric lattice actions, entropy and fundamental groups

Let $ \Gamma $ be a lattice in a noncompact simple Lie Group G, where $ \mathbb{R} - {\rm rank}(G) \geq 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 $ \Gamma $ 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.