14 problems

Let $X$ be a compact cube complex that splits as a graph of virtually special cube complexes with cyclonormal edges. Virtual-specialness conjecture. Then $X$ is virtually special.…

Let $X$ be a compact cube complex that splits as a graph of virtually special cube complexes with cyclonormal edges. A group $G$ has finite stature with respect to a collection of…

Discrete configuration-space conjecture. For all $m\leq n$, the ordered configuration space of $m$ points in $K(n)$ is homotopy equivalent to the $m$th discrete ordered configurati…

Let $k\geqslant 3$, and let $X$ be a finite $\operatorname{CAT}(0)$ cube complex. The boundary distances are the distances between vertices on the boundary of $X$. General reconstr…

Let $k\geqslant 4$, and let $X$ be a finite $\operatorname{CAT}(0)$ cube complex admitting an embedding in $\mathbb{R}^k$. The boundary distances are the distances between vertices…

Let $G\curvearrowright X$ be a virtually special cubulated group, and let a collection of subgroups of $G$ be almost malnormal if distinct subgroups in the collection have finite-i…

Let $G$ act properly, cocompactly, and by isometries on a $\cat$ cube complex. Cubical HHG conjecture. The group $G$ is a hierarchically hyperbolic group (HHG). This would characte…

Cocompact CAT(0) cube complex conjecture. All cocompact CAT(0) cube complexes are hierarchically hyperbolic.

Common finite cover conjecture. There exist finite-sheeted covers

Let $Y$ be a finite nonpositively curved square complex. A square complex is virtually special if it has a finite-sheeted cover that is special. Bridson–Wilton's conjecture. There…

Let an event domain be the domain of an event structure, and call it hyperbolic when its underlying graph is hyperbolic. Hyperbolic event-domain conjecture. Conjectures Thiagarajan…

Let $\mathcal E$ be an event structure with event domain. A directed flat square grid is a directed square grid embedded isometrically in the event domain, with adjacent squares no…

Nonexistence conjecture. There does not exist a maximum complete 1-system $\Gamma$ whose dual cube complex $C(\Gamma)$ is $2$-dimensional.

Virtual specialness conjecture. Virtual specialness is an undecidable property of non-positively curved cube complexes.