21 problems

Let $\Gamma$ be a group with property $(T_n)_{L^1}$, meaning the $L^1$-coefficient form of higher property T used in the source. The $(T_n)_{L^1}$ Zimmer conjecture. Every diffeomo…

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

Let a manifold admit an action of a higher-rank lattice. Margulis–Zimmer conjecture. Under suitable circumstances, the manifold should be homogeneous on an open dense set. The sour…

Topological Zimmer conjecture for closed aspherical manifolds. Every such action factors through a finite group.

Topological Zimmer conjecture for torsion-free subgroups. Every such action factors through a finite group.

Topological Zimmer conjecture. Every such action factors through a finite group; equivalently, every homomorphism

Let $L \backslash H$ be a homogeneous space admitting an action by left multiplications of a simple non-compact Lie group $G$, and suppose that $L$ is non-compact. Zimmer's conject…

Fully supported measure-preserving Zimmer conjecture. Every homomorphism

Zimmer's conjecture. Either $M=S^{n-1}$, or $M=\mathbb{R}P^{n-1}$ and $\alpha$ is $C^{\infty}$-conjugate to the projective action on $S^{n-1}$ or $\mathbb{R}P^{n-1}$.

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…

Homological non-triviality conjecture. When $n>k+1$, any non-trivial action of $\mathrm{SAut}(F_n)$ or $\mathrm{SL}_n(\mathbb{Z})$ on $M^k$ by homeomorphisms is homologically non-t…

Let $G$ be a higher-rank semisimple Lie group, let $\Gamma$ be a higher-rank lattice in $G$, and let $r(G)$ be the root-theoretic dimension defined in the source. Let $M$ be a mani…

Let $G$ be a semisimple group whose simple factors have real rank at least $2$, let $\Gamma$ be a lattice in $G$, and let $\Gamma$ act faithfully and preserving volume on a compact…

Let $G$ be a semisimple Lie group all of whose simple factors have property $(T)$, and let $\Gamma<G$ be a lattice. Assume that $G$ or $\Gamma$ acts smoothly on a compact manifold…

Let $D$ be a semisimple Lie group all of whose factors have property $(T)$, or a lattice in such a Lie group. A rigid geometric structure is a geometric structure of the rigid type…

Let $G$ be a semisimple Lie group with property $(T)$, or let $\Gamma$ be a lattice in such a group, acting smoothly on a compact manifold $M$ as in the source. The measurable-to-s…

Let $\Gamma$ be a cocompact lattice in $Sp(1,n)$ or $F_4^{-20}$, and let $\Gamma'$ be a Gromov-hyperbolic quotient of $\Gamma$. Write $\partial\Gamma'$ and $\partial\Gamma$ for the…

Let $\Gamma$ be a higher-rank lattice acting continuously on the circle $S^1$. The circle-action conjecture. Every such action should be finite. This is part of the low-dimensional…

Let $G$ be a simple Lie group of real rank at least two acting faithfully and preserving volume on a compact manifold $M$. The topological fundamental-group conjecture. The group…

Let $\Gamma<SL(n,\mathbb R)$ be a lattice. An action of $\Gamma$ on a compact manifold is affine-connection preserving when it preserves both a volume form and an affine connection…

Let $n>2$, let $M$ be a manifold, and let $\rho:SL(n,\mathbb Z)\to\operatorname{Diff}(M)$ be a homomorphism. Zimmer's conjecture. The image of $\rho$ is finite when $\dim(M)<n-1$;…