129 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 $G$ 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 $H$ be a coherent hyperbolic group, and let $G$ be an extension of $H$ by $\mathbb{Z}$. Wise's coherence conjecture. The group $G$ is coherent. The source attributes this conje…

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

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…

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…

Rivin's conjecture. The conjugacy growth series $\mathcal P(z)$ of $G$ is rational if and only if $G$ is virtually cyclic.

Let $G$ be a hyperbolic one-relator group. Wise's conjecture. Every such group is virtually free-by-cyclic. If true, this would imply that every hyperbolic one-relator group admits…

A finite-volume hyperbolic manifold is a manifold equipped with a complete finite-volume hyperbolic metric; its fundamental group is cubulated if it acts properly and cocompactly o…

Let $\Gamma\subset \mathsf{PSL}(d,\mathbb R)$ be a subgroup that is Borel Anosov, meaning that it is $P$-Anosov for a Borel subgroup $P$. Sambarino's conjecture. If $\Gamma$ is Bor…

Torsion-free hyperbolic twisted-conjugacy conjecture. The result for free groups should generalize to torsion-free hyperbolic groups and their automorphisms; in particular, the cor…

Undecidability conjecture. The problem of deciding whether a given hyperbolic group has a finite quotient is undecidable. Equivalently, there exists a hyperbolic group which is not…

Let $G$ be a one-relator group whose presentation complex has negative immersions. One-relator hyperbolization conjecture. The group $G$ is hyperbolic. The conjecture would imply t…

Bonk–Kleiner's conjecture. The conformal dimension of $\partial\Gamma$ is attained by a metric in its conformal gauge.

Flat-rank one subgroup conjecture. Every flat subgroup $H$ of flat-rank $1$ in $G$ has a limit set containing two elements, both fixed by $H$. Moreover, $H_1$ is relatively compact…

Convex-cocompactness conjecture. Then $G$ acts discretely, cocompactly, and isometrically on a convex subset of $\mathbb H^3$ with nonempty totally geodesic boundary.

Let $H$ be a finitely presented freely indecomposable subgroup of a torsion-free word hyperbolic group $G$. The virtual normalizer of $H$ is the subgroup of elements of $G$ that no…

Swarup's Strong Accessibility Conjecture. Decompose $G$ maximally over finite subgroups, then decompose each resulting vertex group maximally over two-ended subgroups, and repeat t…

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 $G$ be a hyperbolic group, let $\partial_{\infty}G$ denote its boundary at infinity, let $\operatorname{asdim} G$ denote its asymptotic dimension, and let…

Pro-dense subgroup conjecture. A pro-dense subgroup of a hyperbolic group cannot be finitely generated.

Let $G$ be a group acting properly and cocompactly by isometries on a $\mathop{\rm CAT}(0)$ space. The $\mathbb{Z}\times\mathbb{Z}$ conjecture. Either $G$ is word hyperbolic or $G$…

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…

Let $G$ be a hyperbolic group and let $H$ be any infinite subgroup of $G$. Growth-rate decrease conjecture. The growth rate decreases when taking a quotient of $G$ over $H$. This c…