12 problems

Let $\Sigma=\Sigma_{g,n}$ be a connected surface with genus $g$ and $n$ punctures, with complexity $\xi(\Sigma)=3g-3+n\geq 10$. Let $C\Sigma$ be its curve complex and let…

Let $G$ be a word-hyperbolic group, and let $\partial G$ denote its boundary. A fixed-point-free involution on $\partial G$ is an involutive self-homeomorphism of $\partial G$ with…

Boundary UFC equivalence conjecture. There is a constant-depth circuit that maps the boundary regions as well if and only if the boundary UFCs are equivalent.

No-isolated-points conjecture. Property (ii) should hold for every non-elementary hyperbolic group, namely, every such group has a horofunction boundary with no isolated points.

Let $G$) be a one-ended word hyperbolic group, and let $\partial G$ denote its Gromov boundary. Mess–Mihalik's conjecture. The boundary $\partial G$ is locally connected. The conje…

Sphere-boundary graphical discreteness conjecture. Hyperbolic groups with boundary the $n$-sphere for $n\geq 4$ are graphically discrete.

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

Mihalik's conjecture. The boundary $\partial S_\Gamma$ is path connected if and only if $\Gamma$ is a join.

Let $G$ be a finitely presented group that is hyperbolic relative to a finite collection $c$ of finitely presented proper subgroups, and let $\partial(G,\mathcal P)$ denote its Bow…

Let $(G,\mathcal{P})$ be a type $F$ relatively hyperbolic group pair, with Bowditch boundary $\partial(G,\mathcal{P})$, and let $\operatorname{cd}(G,\mathcal{P})$ denote the cohomo…

Behrstock–Hagen's full visibility conjecture. The cube complex $\mathcal{X}$ is fully visible.

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