33 problems

Let $G$ act geometrically on two $\operatorname{CAT}(0)$ spaces $Y$ and $Y'$ whose $d_L$-metrics stabilize. Let $\partial_M^{\kappa}Y$ and $\partial_M^{\kappa}Y'$ denote their…

Let $G$ be a CK-admissible group acting geometrically on a Hadamard space $Y$. Let $T_{\infty}$ be the Bass–Serre tree associated to the graph-of-groups structure of $G$, and let…

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

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…

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 $X_2$ 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…

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

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…

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.

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 $X$ be a locally compact geodesically complete $\operatorname{CAT}(0)$ space, and let $G$ be a discrete group acting properly and cocompactly on $X$ by isometries. Assume that…

Let $X$ be a locally compact CAT(0) space with a geometric group action. Write $\partial_T X$ for its Tits boundary. Lytchak's conjecture. Then $\partial_T X$ contains a relatively…

Let $X$ be a locally compact geodesically complete CAT(0) space with Tits diameter $\pi$, and suppose that $X$ supports a geometric group action $\Gamma\curvearrowright X$. If $X$…

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…

Highest abelian subgroup conjecture. Any highest abelian subgroup in $G$ is a Morse quasiflat.

Let $\operatorname{NC}(n)$ be the lattice of non-crossing partitions, and let its orthoscheme complex be its simplicial realisation equipped with the orthoscheme metric obtained by…

Let an Artin group be the group associated to a symmetric matrix $(m_{ij})$ with entries in $\{2,3,\dots\}\cup\{\infty\}$, generated by $s_1,\ldots,s_n$ subject to the braid relati…

Let $G$ be a finitely generated group acting nontrivially on a finite-dimensional $\mathrm{CAT}(0)$ complex. Infinite-order element conjecture. The group $G$ contains an element of…

CAT(0)-assumption claim. The CAT(0) assumption in the previous theorem can be dropped.

Morse characterization. The horofunction $h=b_c$ is Morse if and only if there exists a $\delta$ such that, for every $x\in X$, $(i_{\infty})_x$ lies in the $\delta$-neighborhood o…

Let $G$ be a one-ended group acting geometrically on a CAT(0) space $X$, and let $\partial X$ denote its boundary. Assume that $\partial X$ has no finite cut and that a circle of T…