4 problems

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 $(G,\mathcal{P})$ 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…