7 problems

Let $G$ be a torsion-free group and let $R$ be an integral domain. The trivial units conjecture. $G$ satisfies the trivial units property for $R$, meaning that the only units in th…

Let $K$ be a field and let $G$ be a torsion-free group. A Linnell division $K[G]$-ring is an epic division $K[G]$-ring that is $(N,G)$-free for every subgroup $N$ of $G$. Linnell d…

Let $G$ be a torsion-free group, let $C$ be a finite subset of $G$ containing the identity element, and let $n$ be a positive integer. For a finite subset $X\subseteq G$, write…

Let $\mathbb{F}$ be a field and $G$ a torsion-free group. A unit of the group algebra $\mathbb{F}[G]$ is called trivial when it is a non-zero scalar multiple of a group element. Ka…

Let $G$ be a torsion-free group and let $R$ be an integral domain. A non-trivial zero divisor in the group ring $R[G]$ is a nonzero element $\alpha\in R[G]$ for which there exists…

Let $A$ be a finite subset of a torsion-free group, with $|A|\geq 4$. A progression is a finite geometric progression in the group. Freiman's conjecture. If … then $A$ is covered b…

Let $G$) be a group without $R$-torsion, where $R$ is a commutative ring with a unit, and let $k$ be a nonnegative integer. Jump cohomology conjecture. $G$ has jump cohomology of h…