Zimmer's low-dimensional action conjecture

Let $n>2$, let $M$ be a manifold, and let $\rho:SL(n,\mathbb Z)\to\operatorname{Diff}(M)$ be a homomorphism. Zimmer's conjecture. The image of $\rho$ is finite when $\dim(M)<n-1$; the same should hold for any lattice $\Gamma$ in $SL(n,\mathbb R)$. The bound is sharp because $SL(n,\mathbb R)$ acts on $\mathbb P^{n-1}$.