Fisher–Melnick classification conjecture for lattice actions on closed n-manifolds

Let $\Gamma<\mathrm{SL}(n,\mathbb{R})$ be a lattice and let $M$ be a compact manifold of dimension $n$. A smooth action is a homomorphism $\rho:\Gamma\rightarrow\operatorname{Diff}(M)$. The action $\rho$ is said to extend if it extends to an action of $\mathrm{SL}(n,\mathbb{R})$ or its universal cover $\widetilde{\mathrm{SL}(n,\mathbb{R})}$.

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

1. $\rho$ extends to an action of $\mathrm{SL}(n,\mathbb{R})$ or $\widetilde{\mathrm{SL}(n,\mathbb{R})}$;
2. $\rho$ factors through a finite quotient of $\Gamma$; or
3. $\rho$ is built from tori, $G$-tubes, $G$-disks, blow-ups, and two-sided blow-ups, with $\Gamma$ a finite-index subgroup of $\mathrm{SL}(n,\mathbb{Z})$.

The conjecture seeks a complete classification of lattice actions in dimension $n$, extending the known classification of real-analytic and smooth $\mathrm{SL}(n,\mathbb{R})$-actions on closed $n$-manifolds. The classification of lattice actions remains open, despite known exotic constructions such as the Katok–Lewis examples and Fisher–Melnick's examples.