14 problems

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

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…

Osin's conjecture. Acylindrically hyperbolic groups should yield prime factors.

Center conjecture. The center of every irreducible Artin group of infinite type is trivial:

Let $A_\Gamma$ be an Artin group. A standard parabolic subgroup is a subgroup generated by the standard generators associated to a subgraph of the presentation graph, and a subgrou…

Let $A_\Gamma$ be an irreducible Artin group, and let $Z(A_\Gamma)$ denote its centre. The central quotient is $A_\Gamma/Z(A_\Gamma)$. Acylindrical Hyperbolicity Conjecture. The gr…

A group is cubulable if it acts properly and cocompactly on a median graph, and it is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic sp…

Let $G$ be a group acting geometrically on a CAT(0) cube complex. A group is cyclically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space with a lo…

Let $A_\Gamma$ be an irreducible Artin group, and let $Z(A_\Gamma)$ denote its center. Its central quotient is $A_\Gamma/Z(A_\Gamma)$. Acylindrical hyperbolicity conjecture. The gr…

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…

Centre conjecture for non-spherical Artin groups. The centre $Z(A_{\Gamma})$ is trivial:

Artin groups' geometric conjectures. For every Artin group $A_{\Gamma}$: (1) $A_{\Gamma}$ is CAT(0); and (2) the central quotient $A_{\Gamma}/Z(A_{\Gamma})$ is acylindrically hyper…

Let $A_\Gamma$ be an irreducible Artin–Tits group, meaning that its defining graph $\Gamma$ does not decompose as a join of two non-empty subgraphs such that all edges between them…

Let $G$ be a virtually cocompact special group. An acylindrically hyperbolic automorphism-group conjecture. If $G$ is acylindrically hyperbolic, then its automorphism group … is ac…