16 problems

Let $A$) be an Artin group, with corresponding Coxeter group $W$. The standard parabolic subgroups of $A$ of finite type are the parabolic subgroups associated with finite standard…

Kleinian realisation conjecture. The group $G$ is isomorphic to a discrete subgroup of $PSL_2(\mathbb C)$, and $\mathbf S$ is the collection of peripheral subgroups.

Relative hyperbolicity conjecture. The group $G$ is hyperbolic relative to $\mathbf S$.

Let $(G,P)$ be a relatively hyperbolic group pair with $P \cong \mathbb{Z}^2$, and suppose that its Bowditch boundary is homeomorphic to $S^2$. Toral Relative Cannon Conjecture. Th…

Let $G$ be a non-elementary relatively hyperbolic group, and let $\mathcal L(G)$ be its group von Neumann algebra. A von Neumann algebra $\mathcal M$ is s-prime if, for every proje…

Let $G$ be a group that is hyperbolic relative to a finite family of subgroups $\{H_1,\ldots,H_n\}$. Denote by $\mathcal M=\mathcal L(G)$ the von Neumann algebra corresponding to…

Let $\mathcal{G}_{n,p}$ denote the Erdős–Rényi random graph, let $\Gamma\sim\mathcal{G}_{n,p}$, and let $W_{\Gamma}$ be the right-angled Coxeter group with presentation graph…

Finite abelian hierarchy conjecture. Then $G$ should have the $L^2$-Hall property and satisfy the Geometric Hanna Neumann conjecture.

Let $G$ be a relatively hyperbolic diagram group. Free-product conjecture. Relatively hyperbolic diagram groups are free products. The claim gives a more precise form of the preced…

Let $G$ be a relatively hyperbolic diagram group, meaning that $G$ is relatively hyperbolic with respect to a specified collection of peripheral subgroups. Relative-hyperbolicity c…

Let $\Gamma$ be a group relatively hyperbolic relative to a collection $\mathcal{P}$, and let $\rho: \Gamma \to \operatorname{GL}(d,\mathbb{R})$ be a representation satisfying the…

Relative hyperbolicity classification conjecture. $d4a2$ is relatively hyperbolic if and only if this HHS structure has isolated orthogonality.

Let $F$ be a free group, let $w\in F\setminus\{1\}$ have primitivity rank $\pi(w)=2$, and let $G=F/\langle\!\langle w\rangle\!\rangle$. Let $P<G$ be the peripheral subgroup supplie…

A strictly atoroidal Poincaré duality pair of dimension $3$ is the type of pair considered in Wall's conjecture; the supplied passage does not define the terminology further. Wall'…

Let $G$ be a group, let $w\in G*\langle t\rangle$ be unimodular, and let $\widetilde G$ be the group defined by the relative presentation $(*)$ with exponent $k=2$. The preceding t…

Floyd-boundary characterization conjecture. The non-triviality of the Floyd boundary of $G$ should imply that $G$ is relatively hyperbolic.