15 problems

Let $G$ be a finitely generated metabelian group, so its commutator subgroup $G'$ is Abelian, and let $S(G)$ denote the character sphere. For each $n$, write $\Sigma^n(G)^c$ for th…

Let $G$ be a finitely presented self-similar metabelian group, and let $\mathbb{Z} \wr \mathbb{Z}$ denote the wreath product of two copies of the infinite cyclic group. The authors…

Let $A$, $Q$, and $X$ be as in Theorem 1.4: $A$ and $Q$ are finitely generated abelian groups, and $X$ is a set on which $Q$ acts with only finitely many orbits. Regard…

Let $G$ be a uniquely 2-divisible group, and let $(G,\circ)$ be the commutative loop arising from the construction in which $G$ is equipped with the operation $\circ$. A group is m…

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 $A \to G \to Q$ be a short exact sequence of groups, with $G$ finitely generated and $A$ and $Q$ abelian. Regard $A$ as a right $\mathbb{Z}Q$-module via conjugation. Writing…

Subset sum NP-completeness conjecture. The subset sum problem ${\mathbf{SSP}}(G)$ is ${\mathbf{NP}}$-complete.

Jordan-block localization conjecture. The group $M\rtimes C$ should have $H\mathbb Z$-length

The upper-bound conjecture. The group $M\rtimes C$ should satisfy

Let $\Phi_r$ be the free metabelian group on two generators, and let $\Phi_{r,n}$ denote the corresponding finite quotient modulo the relevant power-$n$ congruence subgroup. Using…

The $FP_m$-conjecture for metabelian groups. $\Gamma$ is of type $FP_n$ if and only if every $n$-point subset of $\Sigma^{0}(G;A)^c$ lies in an open hemisphere of…

Let $K$ be an oriented knot, let $p$ be a prime, and let $n\in\mathbb{N}$. Define the $n$-th cyclotomic polynomial by … Let $A_{p,n}=\mathbb{F}_p[t^{\pm1}]/(\phi_n(t))$, with the a…

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.

Let $1 \to A \to G \to Q \to 1$ be a short exact sequence of groups, with $G$ finitely generated and $A$ and $Q$ abelian. For a finitely generated abelian group $Q$, write $S(Q)$ f…

Let $K$ be a knot with knot group $G(K)$, and suppose there is a homomorphism $f:G(K)\twoheadrightarrow M(n\mid p,k)$. Let $N=\langle s\rangle=\{1,s,s^2,\ldots,s^{n-1}\}$ and let…