20 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:

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

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 $G$ be a semisimple Lie group with property $(T)$ and let $\Gamma$ be a lattice in $G$. Let $d$ be the least dimension of an infinite-image linear representation and $n$ the le…

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 $G$ be a semisimple Lie group with property $(T)$ and let $\Gamma$ be a lattice in $G$. Define $d$ as the least dimension of an infinite-image linear representation of $\Gamma$…

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 $\dim(M)<n-1$;…