38 problems

Let $n\leq m$ be positive integers, and let $H$ be a subgroup of the direct product … where each $G_i$ is of type $F_n$ for $1\leq i\leq m$. Say that $H$ is virtually surjective on…

Homological finiteness conjecture. The homological version of the results in the cited work should hold; consequently, homological versions of the two stated theorems on flag-compl…

Naidu–Rowell's conjecture. The category $\mathbf{C}$ has property F precisely when $\operatorname{FPdim}(\mathbf{C})$ is an integer.

Let $G_1,\dots,G_n$ be limit groups and let $\Gamma\subset G_1\times\cdots\times G_n$ be a subgroup. The relevant stronger result replaces “elementarily free groups” by “limit grou…

Conjecture C. The finiteness length satisfies

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…

Higher finiteness-property conjecture. $SV_G$ is of type $\operatorname{\mathcal{F}}_n$ if and only if $G\curvearrowright S$ is of type $[\operatorname{A}_n]$.

Let $G\leq \operatorname{Aut}(\mathcal{T}_d)$ be a contracting self-similar group, and let $V_dG$ be its associated Rӧver–Nekrashevych group. Nekrashevych's conjecture. The group…

Let $G$ be a group acting faithfully on a set $S$, and let $SV_G$ be the associated twisted Brin–Thompson group. For $n\geq 1$, an action is of type $\operatorname{F}_n$ when the r…

Let $n\geq 2$, and let $G$ be strongly shift-similar, so that $H_1(G)=G$. Assume that $G$ has stronger finiteness properties than the classical Houghton group $H_n$. Stronger-finit…

Let $n$ be a positive integer, let $G\leq \operatorname{Sym}(\mathbb{N})$ be shift-similar, and let $\operatorname{F}_{n-1}$ denote the corresponding finiteness property. Finitenes…

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…

Bieri's $\Sigma^m$-conjecture. Every metabelian group of type $\operatorname{F}_m$ has the Bieri $\Sigma^m$-property. This would give a polyhedral description of the BNSR-invariant…

Let $N_1 \to G_1 \to Q$ and $N_2 \to G_2 \to Q$ be short exact sequences. Assume that $Q$ is of type $F_{n+2}$, $G_1$ and $G_2$ are groups of type $F_{n+1}$, and $N_1$ is of type…

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…

A group is non-virtually cyclic if it is not virtually cyclic, and it is of type $\underline{\underline{\operatorname{FP}}}_\infty$ when the trivial Bredon module admits a projecti…

Leary–Nucinkis conjecture. The following conditions are equivalent:

Leary–Nucinkis conjecture. Under these hypotheses, $G$ is of type

Centraliser finiteness conjecture. If the groups $V_r(\Sigma)$ are of type $\operatorname{F}_\infty$, then the centraliser $C_{V_r(\Sigma)}(Q)$ is also of type…

The braided Ptolemy–Thompson groups $T^\sharp$ and $T^*$ are finitely presented, and $T^*$ is asynchronously combable. A group is of type $\operatorname{F}_n$ if it admits a classi…