17 problems

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…

A group is a $PD_n$ group if its fundamental-group cohomology satisfies Poincaré duality over $\mathbb Z$ in dimension $n$. Wall Conjecture. Conversely, every $PD_n$ group is the f…

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…

Kleinian realisation conjecture. The group $G$ is isomorphic to a discrete subgroup of $PSL_2(\mathbb C)$, and $\mathbf S$ is the collection of peripheral subgroups.

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

Kropholler's geometrisation conjecture. The group $G$ is isomorphic to a discrete subgroup of $PSL_2(\mathbb C)$, and $\mathbf S$ is the collection of peripheral subgroups.

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 $g^{-1}H_rg=H_r$.

Loop theorem conjecture. There exists a simple loop $u$ in $F$ which is nullhomotopic in $Y$ but not in $F$.

A 3-dimensional Poincaré duality group is a group satisfying Poincaré duality in dimension $3$, while a 3-manifold group is the fundamental group of a compact 3-manifold. Manifold-…

Let a group with $\operatorname{PD}$ over $\mathbb{Z}$ mean a group satisfying Poincaré duality over the integers, and consider such duality groups in dimension $3$. Dimension-thre…

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…

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

Realization conjecture. Every finitely presented $PD^n$ group is the fundamental group of a closed aspherical manifold.

A strictly atoroidal Poincaré duality pair of dimension $3$ is the type of pair considered in Wall's conjecture; the supplied passage does not define the terminology further. Wall'…

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…