62 problems

Let $G$ be a finitely generated group. Recall that $G$ is virtually free if it has a finite-index free subgroup. Ballier and Stein's conjecture. The group $G$ has decidable domino…

Let $G$ be a hyperbolic group whose boundary $\partial G$ is homeomorphic to $S^{n-1}$. An aspherical closed manifold is a closed manifold whose universal cover is contractible. Gr…

Let $G$ and $G'$ be groups with Cayley graphs whose asymptotic cones are isometric. A group is non-elementary hyperbolic if it is hyperbolic and is not virtually cyclic. Sapir's co…

Let $G:=\pi_1({\mathcal P})$ be the fundamental group of a $2$-dimensional negatively curved acute polygon ${\mathcal P}$ of finite groups. Theorem asserts that when $n=2k$ is even…

Projectivity conjecture. $H$ is projective.

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

Let $G$ be a geometrically rigid hyperbolic group. An abstract commensurator of $G$ is an isomorphism $\phi:H_1\to H_2$ between finite-index subgroups $H_1,H_2\leq G$. A map…

Peterson–Thom conjecture for hyperbolic groups. If $\mathcal{A}\capmathcal{B}$ is diffuse, then the von Neumann algebra they generate is amenable:

Let $G$ be a hyperbolic group with boundary $\partial_\infty G$ homeomorphic to $S^2$, and suppose that $Z^\pm\subset\partial_\infty G$ is a minimal $G$-zipper. Let…

Let $G$ be a hyperbolic group, and suppose that $G$ has property (LR). Let $H$ be a finite index supergroup of $G$. The commensurability conjecture for hyperbolic groups. The group…

Let $G$ be a finitely generated group. For rational $\lambda\geq 1$, real $\epsilon\geq 0$, and a finite generating set $S$, consider the language of $(\lambda,\epsilon)$-quasigeode…

Let $G$ be a Gromov hyperbolic group whose boundary at infinity is homeomorphic to the two-sphere $\mathbb{S}^2$. A group is virtually Kleinian if, up to a finite-index subgroup, i…

Let $\mathbb{F}$ be a finitely generated free group and let $\psi\to \mathbb{F}$ be a fully irreducible monomorphism, meaning that no proper nontrivial finitely generated free fact…

Let $\Gamma$ be a non-elementary hyperbolic group generated by a finite set $S$, and let $\nu$ be a probability measure on $S$ with $\operatorname{supp}(\nu)=S$. Consider the branc…

Let $G$ be a torsionfree hyperbolic group. The torsionfree Cannon conjecture. If the boundary of $G$ is homeomorphic to $S^2$, then $G$ is the fundamental group of a closed hyperbo…

Finite-height subgroup conjecture. If $H$ has finite height in $G$, then $H$ is quasi-convex in $G$.

Let $(G,P)$ be a relatively hyperbolic group pair, meaning that $G$ is relatively hyperbolic relative to the collection of subgroups $P$. Assume that its Bowditch boundary is homeo…

Existence conjecture. There exist an orthogonal representation $\rho:\Gamma\to \operatorname{End}(H)$ weakly equivalent to the regular representation $\lambda_\Gamma$, and a…

Let $G$ be a non-elementary hyperbolic group. A subspace has subexponential growth if its growth function is subexponential, and $G$ is coarsely separable by subspaces of subexpone…

A group is locally quasiconvex if every finitely generated subgroup is quasiconvex, and a group is hyperbolic in the sense of Gromov. A fully irreducible partial endomorphism is a…

Spectral conjecture. There exists such a function $f$ and such representations and vectors for every compact $K\subset\Gamma$ and every $\varepsilon>0$. This would yield, by a limi…

Sphere-boundary graphical discreteness conjecture. Hyperbolic groups with boundary the $n$-sphere for $n\geq 4$ are graphically discrete.

Free-by-cyclic graphical discreteness conjecture. A one-ended, hyperbolic free-by-cyclic group $F_n\rtimes_{\phi}\mathbb{Z}$ with trivial JSJ decomposition is graphically discrete.

Hyperbolic k-geodeticity conjecture. A hyperbolic group $G$ is $k$-geodetic for some positive integer $k$ if and only if $G$ is geodetic. Furthermore, if $G$ is infinite and…

Negative surface curvature implies hyperbolicity. If $\sigma_+(X)<0$ then $\pi_1(X)$ is word-hyperbolic.