Zimmer's conjecture for higher-rank lattice actions

Let $G$ be a connected semisimple Lie group with finite center, all of whose simple factors have real rank at least $2$. Let $\Gamma\subset G$ be a lattice, $M$ be a compact manifold, and $\alpha:\Gamma\to\operatorname{Diff}(M)$ be an action. For a connected Lie group $G$, let $v(G)$ denote the minimal dimension of compact manifolds admitting nontrivial $G$-actions, and let $G_{\mathrm{cpt}}$ be the compact real form of the complexification of the adjoint group of $G$. Zimmer's conjecture. The following statements hold: (1) If $\dim(M)<v(G)$, then $\alpha$ preserves a Riemannian metric on $M$. (2) If $\dim(M)<\min\{v(G),v(G_{\mathrm{cpt}})\}$, then the image $\alpha(\Gamma)$ is finite. This conjecture extends Margulis-type rigidity from linear representations to nonlinear actions. The volume-preserving and non-split cases are central topics of the paper; the conjecture is proved for several families but remains open in general.