The topological fundamental-group conjecture for higher-rank actions

Let $G$ be a simple Lie group of real rank at least two acting faithfully and preserving volume on a compact manifold $M$. The topological fundamental-group conjecture. The group $\pi_1(M)$ should have a finite-index subgroup $\Lambda$ that surjects onto an arithmetic lattice in a Lie group $H$ locally containing $G$. This conjecture seeks topological restrictions on manifolds admitting faithful volume-preserving higher-rank actions; only very special cases are known.