Geometric lattice actions, entropy and fundamental groups

Abstract.

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.