878 problems

Let $G$ be a hyperbolic group whose boundary is homeomorphic to $S^2$. Cannon's conjecture. The group $G$ is virtually Kleinian. This is a central open question in geometric group…

Let $F_n=F(x_1,\dots,x_n)$ be the free group on generators $x_1,\dots,x_n$, and let $\langle x_1,\dots,x_n\mid r_1,\dots,r_n\rangle$ be a balanced presentation, with Andrews–Curtis…

Let $\Gamma$ be a group acting properly discontinuously and cocompactly on $\mathbb{R}^n$ by affine transformations. Auslander's conjecture. Then $\Gamma$ is virtually solvable. Th…

Let $G$ be a Gromov hyperbolic group, and suppose that its Gromov boundary is homeomorphic to the standard Sierpiński carpet. A round carpet is a Sierpiński carpet realized in the…

Let $G=\langle X\rangle$ be a group generated by a finite set $X$, with growth function … Write $f\prec g$ for the growth comparison used in the source: there is a constant $C>0$ s…

Let $Y$ be an aspherical finite $2$-dimensional $CW$-complex, and let $X$ be a connected $CW$-subcomplex of $Y$. Whitehead conjecture. The subcomplex $X$ is aspherical. This conjec…

Let $S=S_{g,r}^n$ be an orientable surface of genus $g\ge 3$ with $r$ punctures and $n$ boundary components, and let…

Eilenberg–Ganea conjecture. For every group $G$,

Valette's conjecture. Property (RD) holds for uniform lattices in $\operatorname{SL}_d(\operatorname{\mathbb{R}})$; more generally, it holds for uniform lattices in any semisimple…

Let $M$ be a closed positive scalar curvature $n$-manifold, and let $\widetilde M$ be its universal covering equipped with the metric lifted from $M$. The macroscopic dimension…

Let $G$ be a Gromov hyperbolic group, and consider affine actions of $G$ on a Hilbert space whose linear parts are uniformly bounded representations. An affine action is proper whe…

A hyperbolic group is a group that is Gromov-hyperbolic with respect to a word metric; a group is residually finite if every nontrivial element survives in some finite quotient. Re…

Ballmann–Buyalo's conjecture. If $X$ contains a geodesic of rank $1$, then it also contains a $G$-periodic geodesic of rank $1$.

Coarse Menger conjecture. There is a function $g:\mathbb{N}^2\to\mathbb{N}$ such that for every $k,d\in\mathbb{N}$, every graph or geodesic metric space $G$, and every two sets…

Brady–McCammond conjecture. The metric space

Let $\Gamma<\mathrm{SL}_N(\mathbb{R})$ be a discrete subgroup, and let the random walk on $\Gamma$ have finite support. Its hitting measure is a measure on the boundary at infinity…

Quasi-isometry classification conjecture. The groups $W_\Gamma$ and $W_{\Gamma'}$ are quasi-isometric if and only if they are isomorphic.

Homogeneity Conjecture. The manifold $(M',ds'^2)$ is homogeneous if and only if every $\gamma\in\Gamma$ is an isometry of constant displacement on $(M,ds^2)$.

Kropholler's conjecture. If there is a proper $H$-almost invariant set $A$ such that $A=AH$, then $G$ has a non-trivial action on a tree in which $H$ fixes a vertex $v$ and every e…

Geoghegan's conjecture. In 1979, Geoghegan conjectured that $F$ has all four properties: it is of type $\mathrm{F}_\infty$, it has no non-abelian free subgroups, it is not amenable…

Hyperbolicity conjecture. Every such group $G$ is hyperbolic.

Let $G$ be a one-relator group, and let $BS(1,n)=\langle a,t\mid ta t^{-1}=a^n\rangle$ denote the Baumslag–Solitar group for an integer $n\neq 0$. Gersten's conjecture. If $G$ cont…

Let $G$ be a finitely generated group. The Ballier–Stein domino problem conjecture. The domino problem for $G$ is decidable if and only if $G$ is virtually free. This conjecture pr…

Juan-Pineda–Leary conjecture. A classifying space ${\mathcal E}_{VC}(G)$ cannot be finite unless $G$ is virtually cyclic.

Thurston's conjecture. The Dehn function of $\Gamma$ is quadratic: