65 problems

Let $\mathsf{\Gamma}$ be a finite connected graph, and let $\mathbb{B}_n(\mathsf{\Gamma})=\pi_1(UC_n(\mathsf{\Gamma}))$ denote its graph $n$-braid group. A right-angled Artin group…

Noetherianity conjecture. The set

Finite-index conjecture. There exists a family $\mathcal{K}$ of cyclic subgroups of $A_\Gamma$ such that $\mathcal{U}(A_\Gamma;\{H\})$ is a subgroup of finite index in…

Let $\mathrm{PFB}_n$ denote the pure flat braid group on $n$ strands. Pure flat braid group conjecture. For every $n \geq 3$, $\mathrm{PFB}_n$ is a right-angled Artin group. This c…

Let $\Gamma=(\mathcal{V},\mathcal{E})$ be a special digraph, let $G$ be the oriented pro-$p$ right-angled Artin group associated to $\Gamma$, and let $q$ be a $p$-power with…

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 Gromov conjecture. Its ma…

Let $A$ be the right-angled Artin group for which Bridson constructed a finitely presented subgroup of $A\times A$ with undecidable conjugacy and membership problems. A right-angle…

Let $\pi$ be a nonabelian surface group with standard generating set … let $X$ be a finite graph, and let … be an embedding. Optimal embedding conjecture. There exists some…

Let $G$ be the outer automorphism group of a finitely generated right-angled Artin group, with virtual cohomological dimension ${\mathrm{vcd}} G=d\geq 2$. A poly-$\mathbb Z$ subgro…

Let $G$ be a diagram group, let $\mathbb{Z} \wr \mathbb{Z}$ denote the lamplighter group, and let a right-angled Artin group mean a group defined by a finite graph presentation wit…

Pure twin group conjecture. $PT_n$ is a right-angled Artin group for each $n\geq 3$.

Let $A_\Gamma$ be a right-angled Artin group associated to a finite simple graph $\Gamma$. For a second homology class $\alpha\in H_2(A_\Gamma)$, let $M_\alpha$ denote the associat…

Let $\mathbb G(\Gamma)$ and $\mathbb G(\Delta)$ be right-angled Artin groups, and let $\operatorname{Core}(\Gamma)$ denote the core graph. RAAG core classification conjecture. If ……

Let $G=\Gamma(\Gamma,\{\mathbb Z\})$ and $G'=\Gamma(\Gamma',\{\mathbb Z\})$ be right-angled Artin groups. Core conjecture. Then … if and only if the cores of $G$ and $G'$ are isomo…

Let $\Gamma$ be a special graph, let $G_\Gamma$ be its associated oriented right-angled Artin group, and let $\mathbb F_2(\Gamma)$ be the quadratic $\mathbb F_2$-algebra generated…

Belk–Bleak–Matucci obstruction conjecture. The group $\mathbb{Z}^d*\mathbb{Z}$ is the correct obstruction for embeddability of right-angled Artin groups in $(d-1)V$.

Let $A=A(\Gamma)$ be an Artin group, let $\mathbb{S}$ be the set of full subgraphs $\Sigma$ of $V(\Gamma)$ such that $A(\Sigma)$ is an irreducible spherical Artin group, and let…

Conjecture that Bridson's group is not coherent. The right-angled Artin group $A$ in Bridson's example cannot be coherent.

Let $A$ be an Artin group associated to a Coxeter system $(W,S)$. Let $\mathcal S$ be the set of irreducible, spherical subsets of $S$, and let $\operatorname{RA}$ be the right-ang…

Pure twin group conjecture. The group $PT_n$ is a right-angled Artin group for each $n\geq 3$.

$R_\infty$ conjecture. Every non-abelian right-angled Artin group has the $R_\infty$-property.

Let $p$ be a prime, let $K$ be a field containing a primitive $p$th root of unity, and let $G_K(p)$ be its maximal pro-$p$ Galois group. Let $L_3$ denote the path graph with four v…

Let $\Gamma$ be a finite simplicial graph, and let $G=G_{\Gamma}$ be the associated right-angled Artin pro-$p$ group. The graph contains an induced subgraph of type $C_4$ when it h…

Normal RAAG conjecture. The subgroup $N$ is isomorphic to one of the right-angled Artin subgroups of $\operatorname{Mod}(S)$ afforded by the authors' construction. In particular,…

Mihalik's conjecture. The boundary $\partial S_\Gamma$ is path connected if and only if $\Gamma$ is a join.