4 problems

Dimension bound conjecture. The dimension of $M$ must be at least $n$.

Fisher–Melnick classification conjecture. For every smooth action $\rho:\Gamma\rightarrow\operatorname{Diff}(M)$, one of the following holds:

Let $n\geq 3$, let $\Gamma<\operatorname{SL}(n,{\bf R})$ be a lattice, and let $M$ be a compact manifold of dimension $n$. A volume-preserving blow-up or two-sided blow-up is one i…

Let $G$ be a higher-rank simple Lie group, let $\Gamma<G$ be a lattice, let $M$ be a compact manifold, and let $\rho:\Gamma\to\operatorname{Diff}(M)$ be an action. Assume that the…