4 problems

Deficiency-one decidability conjecture. If $|A|-|R|=1$, then the Diophantine problem in $G$ is decidable.

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…

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 $K$ 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…