16 problems

Let $R$ be a finitely generated commutative ring with unity that is infinitely generated as a $\b7\mathbb{Z}$-module. For $n\b7\geq 2$, let $\b7\mathbf{A}_n(R)$ be the corresponding…

The homological $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname{FP}_n$, $G_1$ and $G_2$ are of type $\operatorname{FP}_{n+1}$, and $Q$ is of type…

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_{1}$ is of type $\operatorname F_n$, $G_1$ and $G_2$ are of type $\operatorname F_{n+1}$, and $Q$ is of type $\operatorname F_{n+2}$, then…

Let $2\leqslant k\leqslant m$ be integers. For groups $G_{1},\ldots,G_{m}$, a subgroup $P\leqslant G_{1}\times\cdots\times G_{m}$ virtually surjects on $k$ factors if, for every…

Let $d \in \mathbb{N}$ and let (\Delta_n)_{n \in \mathbb{N}) be a sequence of finite $d$-dimensional buildings whose thickness satisfies … as $n \rightarrow \infty$. For each bui…

Finiteness conjecture. The group $V(G)$ and its commutator subgroup $[V(G),V(G)]$ are of type $\operatorname{F}_\infty$.

Let $G$ be an oligomorphic group of permutations of a set $S$, and let $SV_G$ denote the associated twisted Brin–Thompson group. For $n\in\mathbb{N}$, write $\operatorname{F}_n$ fo…

Let $\Vloop=\Thomp{\mathit{LB}_*}$ and $\Floop=\Thomp{\mathit{PLB}_*}$ be the generalized Thompson groups containing the loop braid groups and pure loop braid groups, respectively.…

Let $\Vmock$ be the generalized Thompson group built from the family of mock symmetric groups. Finiteness conjecture. The group $\Vmock$ is of type $\mathrm{F}_\infty$. This is exp…

Bieri–Groves $FP_m$ conjecture. $G$ is of type $FP_m$ if and only if $A$ is $m$-tame as a $\mathbb{Z}Q$-module.

The $n$-$(n+1)$-$(n+2)$ conjecture. If $N_1$ is of type $\operatorname{F}_n$, both $\Gamma_1$ and $\Gamma_2$ are of type $\operatorname{F}_{n+1}$, and $Q$ is of type…

Finiteness-length conjecture. The group

Let $G$ be a subgroup of a direct product $D$ of smooth curve groups, and let $\phi$ be the associated map to the product of the quotient groups, as in the surrounding construction…

Let $G$ be a finitely generated nilpotent-by-abelian group. Let $N$ be a nilpotent normal subgroup with $Q=G/N$ abelian, and consider the factors of the lower central series of $N$…

Let $G$ be a group. Write $\mathfrak F\mathrm{FP}_\infty$ for the finiteness property relative to the family of finite subgroups, and write $\mathrm{FP}_\infty$ for the ordinary fi…

Let $L_1,\ldots,L_n$ be non-abelian limit groups with $n\geq 2$, let $D=L_1\times\cdots\times L_n$, and let $H\leq D$ be a subdirect product that intersects each factor non-trivial…