11 problems

Let $\Gamma$ be a finitely presented $n$-dimensional Poincaré duality group over $\mathbb Z$. Wall's conjecture. There exists a closed $n$-dimensional manifold $M$ which is…

Relative hyperbolicity conjecture. The group $G$ is hyperbolic relative to $\mathbf S$.

Group-pair splitting conjecture. There is an analogue for group pairs of the preceding theorem on $it{PD}^3$ groups.

Acylindrical vertex-pair conjecture. Either $s_i=1$ (and hence $s_j=1$), or $i=j$ and $g^{-1}H_rg=H_r$.

Let $G$ be a $\mathrm{PD}^n$-group that is RFRS. Here $\beta_i^{(2)}(G;\mathbb K)$ denotes the $i$th $L^2$-Betti number with coefficients in a field $\mathbb K$. Poincaré-duality-g…

Let $G$ be a $3$-dimensional Poincaré duality group. Call $G$ atoroidal if it contains no subgroup isomorphic to $\mathbb Z\oplus\mathbb Z$. Atoroidal hyperbolicity conjecture. Eve…

Let $G$ be a finitely presented Poincaré duality group. ENR homology-manifold realization conjecture. $G$ is the fundamental group of a closed aspherical $\operatorname{ENR}$ homol…

Let $G$ be a Poincaré duality group of dimension $3$. Wall's conjecture. $G$ is the fundamental group of a closed aspherical $3$-manifold. This conjecture asks whether the algebrai…

The proper finite-index subgroup conjecture. Every $PD(3)$ group contains a proper subgroup of finite index.

Let $\Gamma$ be an $n$-dimensional Poincaré duality group, and let $b_i^{(2)}(\Gamma)$ denote its $i$th $\ell^2$-Betti number. Singer conjecture. The $\ell^2$-Betti numbers vanish…

A finitely presented group $G$ is an $n$-dimensional Poincaré duality group when its group cohomology satisfies Poincaré duality in dimension $n$. Poincaré duality groups conjectur…