71 problems

Let $d6f2$ be a Coxeter diagram of type $D_n$ with $n\geq 3$. Construct the $(s_n,s_{n-1})$-subdivision $d6f2'_d6f2$ of the simplicial complex $d6f2_d6f2$ by subdividing each edge…

Let $L$ be a flag complex, let $\mathbb{F}$ be a skew field, and let $\varphi\colon A_L\to\mathbb{Z}$ be an epimorphism. Suppose that $BB_L^\varphi$ is of type $\mathsf{F}$. Minima…

Let $L$ be a flag complex, let $\varphi\colon A_L\to\mathbb{Z}$ be an epimorphism, and let $BB_L^\varphi$ be the associated Artin kernel. Suppose $BB_L^\varphi$ is of type…

Let $X=G/K$ be an irreducible symmetric space of rank at least two, let $F$ be a maximal flat in $X$, and let $p_0\in F$. A geodesic in $F$ through $p_0$ is singular if it has the…

Let $W$ be an infinite Coxeter group, let $Y_W$ be the associated hyperplane complement in $\mathbb{C}\otimes V$ restricted to the Tits cone, and let $A$ be the associated Artin gr…

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…

Let $W$ be a Coxeter group, let $G_W$ be its associated Artin group, and let … be the orbit configuration space, where $I$ is the Tits cone in the real reflection representation…

Let $G$ be a braid group, and let its spherical Deligne complex be the complex associated to the spherical standard parabolic subgroups of $G$, equipped with its natural piecewise…

Let $A_S$ be an Artin group whose Coxeter diagram is a tree. A vertex of the Artin complex $\Delta_S$ is of type $\hat s=S\setminus\{s\}$ if it corresponds to a coset of the form…

Let $A_\Gamma$ be an Artin group with defining graph $\Gamma$, and let $\Delta_{S,S'}$ denote the relative Artin complex spanned by vertices of types indexed by $S'$. Suppose…

Let $A_\Gamma^+$ be the Artin monoid associated with an Artin group $A_\Gamma$, and let $\mathbb Z A_\Gamma^+$ and $\mathbb Z A_\Gamma$ be their monoid and group rings. The inclusi…

Let $\Gamma$ be a non-spherical small Coxeter graph, with associated Artin group $A[\Gamma]$, Coxeter group $W[\Gamma]$, and level-two congruence subgroup $A[\Gamma][2]$. Write…

Let $A$ be an Artin group, and let parabolic subgroups mean the parabolic subgroups of $A$. The parabolic intersection conjecture. The intersection of any two parabolic subgroups i…

Let $A_S$ be an Artin group, and let $Z_S$ denote the centre of $A_S$. Acylindrical hyperbolicity conjecture. The quotient $A_S/Z_S$ is acylindrically hyperbolic. This would provid…

Superstability conjecture. An Artin group is superstable if and only if it is abelian.

Domain conjecture. Every quotient $A/Z(A)$ is a domain.

Strong twist conjecture. If $U\subseteq A$ is an Artin generating set with $R_S=R_U$, then $S$ and $U$ are twist equivalent.

Weak twist conjecture. If $U\subseteq A$ is an Artin generating set with $R_S=R_U$, then $\Gamma_S$ and $\Gamma_U$ are twist equivalent.

Let $A_\Gamma$ be an Artin group and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. Let $\mathrm{Liv}^\chi$ denote the living subgraph associated to $\chi$, and let…

Let $A_\Gamma$ be an Artin group satisfying the $K(\pi,1)$-conjecture, and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. Write $\mathrm{Liv}^\chi$ for the living subgraph, and c…

Let $A_\Gamma$ be an Artin group and let $\chi:A_\Gamma\to\mathbb{R}$ be a character. The living subgraph $\mathrm{Liv}^\chi\subset\Gamma$ is obtained by deleting vertices $v$ with…

Let $W$ be a Coxeter group, let $\delta\in W$ be a Coxeter element, and let ${\normalfont\textsc{Br}}_{W,\delta}^{m}$ be the bisimplicial complex constructed with algebraic cell la…

Let $A[D_5]$ be the Artin group of spherical type $D_5$, and let $Z(A[D_5])$ denote its center. The $D_5$ property $R_\infty$ conjecture. The Artin group $A[D_5]$ and its central q…

Let $A$ be an Artin group such that every irreducible $A_Z$ with $|Z|>2$ is not of spherical type. For $X\subseteq S$, write … If $A_X$ has no cyclic irreducible component…

Fundamentality conjecture. If $\Gamma$ is connected and twistless, then $\Gamma$ is fundamental.