14 problems

Property H conjecture. Every Artin–Tits presentation satisfies Property $H$. The conjecture asks whether transformations of type $\infty$ are unnecessary for reducing every word re…

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 $e004\gamma i G$. This would provide a uniform bound for the centralizers needed by the ca…

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 $A=A_S$ 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 $A=A_S$ be a non-cyclic irreducible Artin–Tits group of spherical type. Let $X_P^A$ be the union of all proper irreducible standard parabolic subgroups of $A$ and the cyclic su…

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…

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…