17 problems

A rational normal form is a normal form used to estimate the minimal length of elements in a braid group, and Dehornoy handle-reduction is a procedure for shortening such represent…

Sibylla's tameness conjecture. Every Garside monoid has a tame Garside element.

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 $G$ be a Garside group, let $x$ be a rigid element, and consider the conjugacy graph of $x^n$. Its vertices have levels, with level $1$ and maximal-level vertices as in the sou…

Let $G$ be a Garside group. For an element $y\in G$ that is conjugate to a rigid element and has a rigid power, let $r(y)$ be the smallest positive integer such that $y^{r(y)}$ is…

Let $G$ be a Garside group equipped with a Garside structure, and let $SC(x^n)$ denote the sliding circuits set of $x^n$. Uniform periodicity conjecture. There exists a finite set…

Let $G$ be a Garside group and let $x\in G$ be rigid. For each positive integer $n$, let $SC(x^n)$ denote the sliding circuits set of $x^n$. Boundedness conjecture. For any rigid e…

Let $M$ be a Garside monoid with two atoms, and let $G(M)$ denote its enveloping group. Picantin's conjecture. The group $G(M)$ is isomorphic to either a group of the form … for po…

Hypersurface-complement conjecture. The group $G$ is the fundamental group of the complement in $\mathbb{C}^3$ of an algebraic hypersurface.

Let $B_n$ be the $n$-strand braid group, and let $\mathcal C_{\rm AL}(B_n)$ be its additional length graph. Let the curve graph of the $(n+1)$-punctured sphere be the graph whose v…

The examples described are Garside groups generated by two elements, including Artin–Tits groups of dihedral type and torus knot groups. Classification conjecture. These are the on…

Weak proper discontinuity conjecture. The action of $x_n$ on $\mathcal C_{AL}(B_n)$ is weakly properly discontinuous.

Braid-group quasi-isometry conjecture. The additional length complex is quasi-isometric to the curve complex:

Infinite diameter conjecture. Apart from obvious counterexamples, additional length complexes have always infinite diameter.

Let $G$ be an Artin group with Coxeter group $W$, and let $g$ be a Coxeter element of $W$. The interval $[1,g]^W$ is the interval in the absolute order determined by $g$, and the c…

A non-degenerate, braided and involutive set-theoretical solution $(X,S)$ consists of a set $X$ with a map $S:X\times X\to X\times X$ having these properties. Such a solution is de…

Let $G$ be a Garside group with Garside element $\Delta$ and canonical length parameter $k$; let $T$, $M$, and $R$ be the quantities defined in the preceding algorithmic complexity…