61 problems

Ballmann–Buyalo's conjecture. If $X$ contains a geodesic of rank $1$, then it also contains a $G$-periodic geodesic of rank $1$.

Let $X$ be an irreducible, geodesically complete, proper $\operatorname{CAT}(0)$ space, and let a geometric action on $X$ be one for which $X$ is neither a higher-rank symmetric sp…

Let $A_\Gamma$ be an infinite-type Artin group, and let $D_\Gamma$ be its Deligne complex, whose vertices are cosets $aA_T$ of spherical-type standard parabolic subgroups and whose…

Let $G$ be an infinite discrete group acting geometrically on an irreducible $\operatorname{CAT}(0)$ space $X$. A geodesic in $X$ is rank one when it does not bound a flat half-pla…

Ballmann's higher rank rigidity conjecture. Any Hadamard space of rank at least two with a geometric group action admits an affine map which is not a dilation.

Let $G$ act geometrically on two proper CAT(0) spaces $X$ and $Y$, and let the corresponding Myrberg sets be the subsets of their respective boundaries consisting of Myrberg points…

Let $G$ act properly and cocompactly by isometries on a CAT(0) space $X$, and let $d$ be a positive integer. Flat Closing Conjecture. The space $X$ contains a $d$-dimensional flat…

Let $X$ be a CAT(0) space with Tits boundary $\partial X$ of diameter $\pi$. Ballmann–Buyalo's classification conjecture. In this remaining case, $X$ is either a symmetric space, a…

CAT(0) conjecture. The universal cover of the $v$-splitting of $W$ determined by $S'$ is $\operatorname{CAT}(0)$.

Let $G$ be a limit group, also called a fully residually free group, and let a group act properly and cocompactly by isometries on a $\mathop{\rm CAT}(0)$ space with the Isolated F…

Let $G$ be a group acting properly and cocompactly by isometries on a $\mathop{\rm CAT}(0)$ space. The $\mathbb{Z}\times\mathbb{Z}$ conjecture. Either $G$ is word hyperbolic or $G$…

Let $\Gamma$ be a Coxeter graph, and let $\mathrm{vD}_{\Gamma}$ be the virtual Deligne complex associated with the virtual Artin group $\mathrm{VA}[\Gamma]$. Equip…

Gromov's dimension conjecture. The topological dimension agrees with the geometric dimension:

Let $\mathfrak{Q}$ be a quandle product with trivial holonomy of finitely many groups, each of which is cocompactly cubulable or CAT(0). Cocompact cubulation and CAT(0) conjecture.…

Let $G$ be a countable group acting geometrically on a CAT(0) complex $X$ of dimension $2$. A flat plane in $X$ is a subspace isometric to $\mathbb{R}^2$. Flat closing conjecture.…

Let $X_2$ be the specific compact locally CAT(0) triangle-square complex constructed in the paper. Biautomaticity conjecture for $X_2$. The fundamental group $\pi_1(X_2)$ is biauto…

Let $X$ be a compact locally CAT(0) triangle-square complex such that no intrinsically aperiodic flat embeds in $\widetilde{X}$ and there are only finitely many distinct thoroughly…

Let $X$ be a compact locally CAT(0) triangle-square complex. A flat is thoroughly crumpled if it is crumpled and there is a bound on the number of cells in each region. Levitt–McCa…

Let $X$ be a compact locally CAT(0) triangle-square complex, and let $F$ be an intrinsically aperiodic flat, meaning that $F$ does not admit a cocompact cellular action of…

Let $G$ be a finitely generated group acting by isometries on a finite-dimensional CAT(0) complex. NOP conjecture. If the action has no global fixed point, then $G$ contains an ele…

Equality conjecture. For every $d\geq 5$,

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 $S$ be a finite set of generators, let $d\in(0,1)$ be a density, and consider a random group in the Gromov density model with density $d$ and relator length tending to infinity…

Torsion conjecture. If $\Gamma$ is torsion, then $\Gamma$ is finite. An easier version is that if $\Gamma$ is almost-cocompact, then it contains an infinite-order element.