18 problems

Let $(G,P)$ be a relatively hyperbolic group pair, meaning that $G$ is relatively hyperbolic relative to the collection of subgroups $P$. Assume that its Bowditch boundary is homeo…

Let $G$ be the fundamental group of a non-sporadic graph of groups with finite edge stabilisers, and let $\Phi\in\operatorname{Aut}(G)$ be fully irreducible relative to this splitt…

Relative biexactness conjecture. $G$ is biexact relative to $\{H_i\}_{i=1}^{n}$.

Let $G$ be a finitely generated group hyperbolic relative to a collection of subgroups $\{H_\mu\}_{\mu \in \Lambda}$ of $G$. Relative bi-exactness conjecture. If all subgroups…

Let $(G,\mathcal{P})$ be a relatively hyperbolic group pair, with Bowditch boundary $\partial(G,\mathcal{P})$. A boundary is planar if it embeds in the plane, and it has no cut poi…

Let $G$ be a finitely generated group hyperbolic relative to a finite collection $\{P_1,\ldots,P_n\}$ of proper finitely generated subgroups. Semistability conjecture. If each…

Let $G$ be a one-ended group and let $(G,\mathcal{P})$ be a relatively hyperbolic pair, with Bowditch boundary $\partial(G,\mathcal{P})$. Assume that this boundary is planar and ha…

Planarity conjecture. If $\partial_B(G,\mathcal{P})$ is planar and has no cut points, then $G$ is virtually isomorphic to a Kleinian group.

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…

Cohomological dimension formula.

The relative Cannon conjecture. If

Aschenbrenner–Friedl–Wilton's conjecture. If $H$ is a fully relatively quasiconvex subgroup of $\pi_1M$, then $H$ is a virtual retract. In particular, $H$ is separable.

Let $G$ be torsion-free and hyperbolic relative to a collection $\mathcal{H}$. Let $\operatorname{cd}(G,\mathcal{H})$ be the cohomological dimension of the pair $(G,\mathcal{H})$,…

Relative quasiconvexity criterion. Then $H$ is relatively quasiconvex in $G$.

Let $N$ be an orientable, compact, non-positively curved $3$-manifold with empty or toroidal boundary, and write $\pi=\pi_1(N)$. Equip $\pi$ with its natural relatively hyperbolic…

NRH subgroup conjecture. If $G$ is hyperbolic relative to a collection of NRH proper subgroups, then $G$ acts geometrically finitely on its Floyd boundary.

Olshanskii–Osin–Sapir conjecture. If $G$ has non-trivial Floyd boundary, then $G$ is hyperbolic relative to a collection of proper subgroups.

Let $G$ be a group hyperbolic relative to the peripheral system $\mathcal{P}=\{P_1,\ldots,P_n\}$. Suppose that hyperbolic groups are residually finite, and that each $P_i$ is finit…