8 problems

Let $G<\operatorname{Homeo}(S^2)$ be a geometrically finite convergence group. Martin–Skora's covering conjecture. Then $G$ is covered by a Kleinian group. Here, being covered mean…

Let a group act on $S^{2}$ as a uniform convergence group action, meaning that the action is a convergence group action and is uniform. A cocompact Klein group is a discrete group…

Let a convergence group be a group acting on $S^2$ as a convergence group. Martin--Skora conjecture. Any convergence group acting on $S^2$ is covered by a Kleinian group. This is a…

Martin–Skora's conjecture. Convergence groups can be realized as Kleinian groups.

Uniform convergence conjecture. This action is a uniform convergence group.

Mapping torus convergence-group conjecture. For any p-A-like element $\phi$ of $G$, the mapping torus subgroup $G_\phi$ acts on $S^2$ as a convergence group.

Fuchsianity conjecture. The group $G$ is a Fuchsian group, equivalently, a convergence subgroup of $\operatorname{Homeo^+}(S^1)$.

Promotion of pseudo-fibering conjecture. There are three possibilities: $G$ is elementary; $G$ is a Fuchsian group; or $G$ is a closed hyperbolic $3$-manifold group.