Volume-preserving classification conjecture for lattice actions

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 in which every vector field used in the construction has derivative $n$ at $0$. Volume-preserving classification conjecture. Any action $\rho:\Gamma\to\operatorname{Diff}(M)$ either factors through a finite quotient of $\Gamma$, or $\Gamma\leq\operatorname{SL}(n,{\bf Z})$ is a finite-index subgroup and the action is built from tori and volume-preserving blow-ups and two-sided blow-ups. This is the volume-preserving specialization of the proposed classification and excludes extensions to $\operatorname{SL}(n,{\bf R})$ or its universal cover under the preceding proposition.