13 problems

Levin conjecture. Every group equation over a torsion-free group $G$ is solvable over $G$.

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 $F$ be an orderable group, let $R={\mathbb Z}F$ be its integral group ring, and write $U(R)$ for the group of units of $R$. The unit conjecture. The units of $R$ are the obviou…

Let $U$ be a torsion-free group, and let $\mathbb{C}[U]$ denote its complex group ring. A projection is an element $e$ satisfying $e=e^*=e^2$; it is nontrivial when $e\ne 0$ and…

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 group and let $\mathbb C G$ be its complex group algebra. An element $\alpha\in\mathbb C G$ is regular if $\alpha\beta\neq 0$ for every nonzero $\beta\in\mathbb C G$.…

Let $G$ be a torsion-free group and let $T$ be an element of its group ring. Let $\mu_T$ be the spectral measure of $T$. Torsion-free singular-continuity conjecture. The spectral m…

Let $G$ be a torsion-free group, and let $\mathbb{Z}G$ be its integral group ring. Trivial unit conjecture. Every invertible element of $\mathbb{Z}G$ is of the form $\pm g$ for som…

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 $K$ be a field and let $G$ be a torsion-free group. A unit in the group algebra $KG$ means a two-sided multiplicative unit. Unit conjecture. Every unit of $KG$ is trivial, name…

Let $G$ be a torsion-free group, and let $k$ be the parameter in the paper's lower-bound theorem for $|AB|$. Equal-set linear-threshold conjecture. When $A=B$, the lower bound on…

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…