29 problems

Let $(G,\mathcal{P})$ be a relatively hyperbolic group pair whose Bowditch boundary is homeomorphic to $S^2$. Relative Cannon conjecture. Then $G$ acts properly, isometrically, and…

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…

Peripheral embedding conjecture. $G$ quasiisometrically embeds into a product of binary trees if and only if each peripheral subgroup $H_t$ does as well.

Let $G$ be a group that is weakly hyperbolic relative to a finite collection of subgroups $\{H_\lambda\}_{\lambda\in\Lambda}$. Assume that each subgroup $H_\lambda$ has finite asym…

Symmetric geodesics conjecture. For any $k\ge 0$, there exists a constant $\varkappa=\varkappa(k)$ such that, whenever $p,q$ are a pair of $k$-similar geodesics in…

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…

Let $G$ be a relatively hyperbolic group whose boundary is a topological Schottky set. Hruska–Walsh conjecture. Such groups are all conjectured to be virtually discrete subgroups o…

Spectral non-degeneracy conjecture. For all sufficiently large $n$, the random walk determined by $\mu_n$ is spectrally non-degenerate.

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 group hyperbolic relative to finitely generated subgroups $H_1,\ldots,H_N$, and suppose that the generalized conjugacy problem (GCP) in each parabolic subgroup $H_i$ c…

Let $G$ be a group and let $(G,\mathcal{P})$ be a relatively hyperbolic pair. Its Bowditch boundary is denoted by $\partial(G,\mathcal{P})$. Assume that $G$ is one ended and that t…

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 $G$ 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…

Cohomological dimension formula.

Let $\Gamma$ be a finitely generated group hyperbolic relative to a collection of subgroups, and let a $Z$-boundary and the Floyd boundary of $\Gamma$ be given as in Corollary. Aut…

Wall-type realization conjecture. If $(G,\mathcal{P})$ is a $\PD(3)$ pair, then $G$ is the fundamental group of an aspherical $3$-manifold with boundary, where $\mathcal{P}$ is the…

Degree-of-commutativity conjecture. If $G$ is not virtually abelian, then

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 $G$.