37 problems

Let $\Gamma$ be a planar graph, and let $B_n\Gamma$ and $P B_n\Gamma$ denote its $n$-strand braid group and pure braid group, respectively. A group is right-angled Artin if it admi…

Rigidity conjecture. If $\Gamma$ has no satellites, then it is rigid.

Noetherianity conjecture. The set

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…

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…

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…

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.

Let $A(\Gamma)$ be a right-angled Artin group whose defining graph $\Gamma$ is connected and not a join, and let $H$ be a stable subgroup of $A(\Gamma)$. Let $(\mu_i)$ be a sequenc…

Let $S_g$ be a closed orientable surface and let $N$ be a normal subgroup of $\operatorname{Mod}(S_g)$ that is isomorphic to a right-angled Artin group. Normal right-angled Artin s…

Let $\Gamma$ be a finite graph, and let $A(\Gamma)$ be the associated right-angled Artin group. Say that a group has MCF word problem when its word problem is a multiple context-fr…

Let $\mathbb{G}(\Gamma_1)$ and $\mathbb{G}(\Gamma_2)$ be $2$-dimensional right-angled Artin groups, meaning that their commutation graphs are triangle-free. Let…

Let $N$ be a normal subgroup of either the orientation-preserving mapping class group $\operatorname{Mod}(S_g)$ or the extended mapping class group $\operatorname{MCG}(S_g)$. Norma…

Non-embeddability conjecture. If $p,q,r$ are large enough, then $G(T_{p,q,r})$ does not embed into $G(P_n)$ for any $n$.

Linear-or-quadratic distortion conjecture. The distortion of $H_\Phi$ in $A_\Gamma$ is linear or quadratic.

Let $\Gamma$ be a finite simplicial graph, and let $A_\Gamma$ be the right-angled Artin group defined by generators corresponding to the vertices of $\Gamma$, with two generators c…

Let $D(\mathcal{P},w)$ be a diagram group, and let $A(\mathcal{P},w)$ denote the associated right-angled Artin group. A diagram group is embeddable into a right-angled Artin group…

Let $G$ be a right-angled Artin group, and let $h(G)$ be the minimum of $b_2(M)$ over all closed, oriented topological $4$-manifolds $M$ with fundamental group isomorphic to $G$. L…

Let $A_\Gamma$ be a right-angled Artin group, and let $\Gamma$ be austere, meaning that its defining graph has no nontrivial graph symmetries, no dominated vertices, and no separat…