10 problems

Let $G$ be a CAT(0) group, meaning a group acting geometrically on a CAT(0) space, and suppose that $G$ contains a rank-one element. Quotient conjecture. $G$ has a proper quotient…

CAT(0) Cannon conjecture. The group $G$ has a finite-index subgroup that is the fundamental group of an aspherical $3$-manifold.

Let $\Gamma$ be an extended presentation graph, and for each edge $e$ let $\mathrm{Sh}_e$ denote the corresponding edge parabolic subgroup. An edge parabolic embeds if the homomorp…

Let $\Gamma$ be a special graph, let $G_\Gamma$ be its associated oriented right-angled Artin group, and let $\mathbb F_2(\Gamma)$ be the quadratic $\mathbb F_2$-algebra generated…

Finite-center conjecture. If $G$ is a CAT(0) group and $G$ and every finite-index subgroup of $G$ have finite center, then $G$ has AIP. This would clarify when the AIP hypothesis u…

Generalized boundary conjecture. The group $G$ acts on a CAT(0) space $X$ such that $\partial X$ is not path connected.

Planar boundary conjecture. Every visual boundary of $G$ is planar, and furthermore, $G$ is virtually the fundamental group of a compact 3-manifold.

Non-rigidity conjecture. The group $G$ will be a non-rigid CAT(0) group with uncountably many boundaries.

Let $X$ be a complete $\mathrm{CAT}(0)$ space and let $G$ be a group acting geometrically on $X$, meaning properly and cocompactly by isometries. A geodesic line in $X$ has rank on…

Main conjecture. Both schmear maps are cell-like; hence $\partial X_{1}$ and $\partial X_{2}$ are $G$-equivariantly cell-like equivalent.