166 problems

Let $A_S$ be an Artin group, and let $S'ubset S$ be almost spherical. Let $\Delta_{S,S'}$ denote the relative Artin complex spanned by vertices of types indexed by $S'$. Assume tha…

Let $A_\Gamma$ be an infinite-type Artin group, and let $D_\Gamma$ be its Deligne complex, whose vertices are cosets $aA_T$ of spherical-type standard parabolic subgroups and whose…

Twist-equivalence conjecture. If $A_\Gamma\cong A_\Lambda$, then $\Gamma$ and $\Lambda$ 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 $A_S$ be a spherical Artin group, and let $\Delta_S$ be its spherical Deligne complex, equipped with its naturally defined metric. Charney–Davis conjecture. The complex…

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…

Non-splitting conjecture. The quotient homomorphism $\operatorname{Aut}(\pa)\to \operatorname{Aut}(\pa/Z)$ has no splitting.

Let defining graphs encode Coxeter groups, and let diagram twist equivalence be the equivalence relation generated by diagram twists over the family of all defining graphs. Diagram…

Consider an Artin–Tits group of spherical type equipped with its classical and dual Garside presentations. A standard parabolic subgroup for the dual presentation is a subgroup gen…

Let $(W,S)$ be a finite-rank Coxeter system, let $R$ be the set of reflections of $W$, and let $l_R(w)$ be the minimum number of reflections needed to express $w$. Fix a Coxeter el…

Tits's conjecture. The only relations in $A$ among the squares of the generators are consequences of the relations

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 $\Gamma$ be a Coxeter graph and let $\mathrm{A}[\Gamma]$ be the associated Artin group. Artin torsion-freeness conjecture. The Artin group $\mathrm{A}[\Gamma]$ is torsion-free…

A Dyer system $(D,X)$ has standard parabolic subgroups $D_Y$ generated by subsets $Y\subseteq X$. For subsets $Y,Y'\subseteq X$, let $\operatorname{Conj}(D_Y,D_{Y'})$ be the set of…

Let $\Gamma$ be a finite simplicial graph with labeling map $l:E(\Gamma)\to\{2,3,4,\dots\}$, let $A_\Gamma$ be its Artin group, and let $\mathcal{P}$ be the poset of spherical spec…

The dual Artin isomorphism conjecture. The canonical epimorphism

Acylindrical-hyperbolicity conjecture for Artin groups. All Artin groups are acylindrically hyperbolic unless they are direct products or spherical.

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 $A$ be an Artin group, and let the canonical complex manifold associated to $A$ be the space under consideration. The $K(\pi,1)$-conjecture. This manifold is aspherical. The co…

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 finite-type Coxeter graph with associated Artin group $A_\Gamma$, Coxeter group $W_\Gamma$, and complexified hyperplane complement $\mathcal H_\Gamma$. The quotie…