23 problems

Contractibility conjecture. The space

Let $G$ be an Artin-Tits group of spherical type, let $\mathcal C(G)$ be its curve graph, and let $\alpha\in G$. Suppose a vertex of $\mathcal C(G)$ has a finite orbit under…

Let $G$ be an Artin-Tits group of spherical type with rank $n$, let $\mathcal C(G)$ be the curve graph, and let $\alpha\in G$. Consider a vertex of $\mathcal C(G)$ with a finite or…

Uniform period bound conjecture. There is a number $n(G)$ such that $p\leq n(G)$ for every $e004gamma i G$. This would provide a uniform bound for the centralizers needed by the ca…

Let $W$ be a Coxeter group acting on its reflection representation, let $\Theta=V^*$, and let $\operatorname{Cone}\subseteq\Theta$ be its Tits cone with interior…

K(π,1) conjecture. The space $\operatorname{Stab}^\dagger_{\mathcal C}(\mathcal D)$ is contractible; consequently, both $\Upsilon_{\operatorname{reg}}/\mathbb W$ and…

Faithfulness conjecture. The action of the Artin–Tits group $\mathbb B$ on the triangulated category $\mathcal D$ is faithful.

Let $A_{\Gamma}$ be an Artin–Tits group with standard generating set $S$, Coxeter graph $\Gamma$, and rank $|S|$. The group is irreducible when $\Gamma$ is connected, and its centr…

Let $A=A_S$ be an irreducible Artin–Tits group of spherical type with rank at least $3$. Let $X_{abs}^A$ be the absorbable-element generating set, $X_{NP}^A$ the union of normalize…

Let $A'$ denote one of the commutator subgroups considered in the paper, including the cases where the commutator subgroup is not covered by the preceding argument, such as…

Let $A$ be an irreducible Artin-Tits group of spherical type acting on the additional length graph $\mathcal{C}_{AL}$, and let $G\leq A$ be a subgroup with generating system $S$. W…

Let $M$ be an Artin–Tits monoid. A padding of a multifraction is obtained by inserting an even number of trivial components at its beginning; semi-convergence up to $f$-padding mea…

Let $M$ be an Artin–Tits monoid, and let $\mathcal{R}$-reduction be the reduction system on multifractions associated with $M$. Conjecture A. $\mathcal{R}$-reduction is semi-conver…

Let $M$ be an Artin–Tits monoid, let $\mathcal{F}_{M}$ be the set of multifractions over $M$, and let $\operatorname{red}_{\mathrm{t}}$ denote tame reduction. A multifraction is un…

Let $M$ be an Artin–Tits monoid and let $\mathcal{R}_{M}$ be its reduction system on multifractions. The system is uniformly cross-confluent if there is a map $\nabla$ from multifr…

Let $M$ be an Artin–Tits monoid and let $\mathcal{R}_{M}$ be its reduction system on multifractions. Two multifractions are right reducts of the same multifraction if they are obta…

Normalizer–quasi-centralizer conjecture. For every Artin–Tits group and every subset $X$ of its standard generating set,

Double-centralizer conjecture. If $A_X$ is not of spherical type, then

Let $\langle S,R\rangle$ be an Artin–Tits presentation, let $w\in S\cup S^{-1}$ be a word, and write $\overline{w}$ for the group element represented by $w$. Let…

$K(\pi,1)$ conjecture. For every Coxeter graph $\Gamma$, the quotient

Conjecture B. If $\Gamma$ is non-spherical and connected, then

Let $(\mathcal{S},\mathcal{R})$ be an Artin–Tits presentation, and let denote the reducing equivalence defined earlier in the paper. Reducing conjecture. Every Artin–Tits presentat…

Left–right ultra summit graph conjecture. For every element $x\in G$, not necessarily rigid, the left ultra summit set graph and the right ultra summit set graph of $x$ are isomorp…