Zimmer's conjecture on minimal dimensions and invariant metrics

Let $G$ be a connected simple Lie group with finite center, let $\Gamma\subset G$ be a lattice, let $M$ be a compact manifold, and let $\mathrm{vol}$ be a volume form on $M$. The quantities $d_{\mathrm{rep}}(G)$, $d_{\mathrm{cpt}}(G)$, and $d_0(G)$ are the dimensions defined in the source.

Zimmer's conjecture. (1) If $\dim(M)<\min\{d_{\mathrm{rep}}(G),d_{\mathrm{cpt}}(G),d_0(G)\}$, every homomorphism $\alpha:\Gamma\to\mathrm{Diff}(M)$ has finite image. (2) If $\dim(M)<\min\{d_{\mathrm{rep}}(G),d_{\mathrm{cpt}}(G)\}$, every homomorphism $\alpha:\Gamma\to\mathrm{Diff}_{\mathrm{vol}}(M)$ has finite image. (3) If $\dim(M)<\min\{d_0(G),d_{\mathrm{rep}}(G)\}$, then $\alpha(\Gamma)$ preserves a Riemannian metric. (4) If $\dim(M)<d_{\mathrm{rep}}(G)$, then every homomorphism $\alpha:\Gamma\to\mathrm{Diff}_{\mathrm{vol}}(M)$ has image preserving a Riemannian metric.

The conjecture generalizes the low-dimensional finiteness assertions for lattices in $\mathrm{SL}(n,\mathbb{R})$. The source records partial results for several non-exceptional split real forms and regularity classes, while the full formulation remains open.