9 problems

Factor-system conjecture. Every cubical group admits a factor system.

Let $A_\Gamma$ be an Artin group, let $C_\Gamma$ be its associated clique-cube complex, and let … be the action by left multiplication. For a subgroup $H\subset A_\Gamma$, write…

A group $G$ has property ${\rm F}\mathcal{C}'$ if every cellular action of $G$ on a finite-dimensional CAT$(0)$ cube complex is locally elliptic. A subgroup is non-trivial if it is…

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…

Roller-boundary conjecture. The action of $\Gamma$ on $\partial_R X$ is meandering-hyperbolic for a suitable metric on $\partial_R X$.

Wise's conjecture. Every torsion-free one-relator group with no subgroup isomorphic to $BS(m,n)$, for $m\neq\pm n$, acts freely on a CAT(0) cube complex.

Free-by-cyclic CAT(0) cube complex action conjecture. Every free-by-cyclic group acts freely on a CAT(0) cube complex.

Let $\Gamma$ be an irreducible lattice in a semisimple Lie group of rank at least $2$, and let $X$ be a CAT(0) cube complex, without assuming that $X$ is finite-dimensional. Fixed-…

Let $P$ be a semigroup presentation, let $w$ be a word in the semigroup presented by $P$, and let $D(P,w)$ be the associated diagram group acting on the proper CAT(0) cube complex…