67 problems

Let $G$ be a finitely generated amenable group. Its integral group ring is denoted by $\mathbb{Z}G$. A group ring is properly coherent when it is coherent but not Noetherian. A pro…

Let $G$ be a group. Its integral group ring is denoted by $\mathbb{Z}G$. Baer's conjecture. $\mathbb{Z}G$ is Noetherian if and only if $G$ is virtually polycyclic. The forward i…

Zassenhaus conjecture. If $u\in\mathcal{U}(\mathbb ZG)$ is a unit of finite order, then $u$ is conjugate within $\mathbb QG$ to $\pm g$ for some $g\in G$.

Matrix-kernel approximation conjecture.

Let $R$ be an integral domain and let $G$ be a torsionfree group. An idempotent in the group ring $RG$ is an element $e$ satisfying $e^2=e$. Kaplansky's conjecture. Every idempoten…

Let $F$ be a field of characteristic zero and $G$ a group. Let $\operatorname{class}_F(G)_f$ denote the indicated $F$-vector space of finite conjugacy classes, and let…

Let $R$ be a commutative integral domain and $G$ a group. For $g\in G$, write $\operatorname{HS}_{RG}(P)(g)$ for the value at the conjugacy class $(g)$ of the Hattori–Stallings ran…

Let $G$ be a group, let $H$ be a subgroup of finite index, and let $R$ be an arbitrary ring. Assume that for every nontrivial element $x$ of $G$, at least one of the following cond…

Moore's conjecture. Assume that for every nontrivial element $x$ in $\Gamma$, at least one of the following conditions holds: $M1)$ $\langle x\rangle\cap H\neq\\{e\\}$; or $M2)$…

Isomorphism Conjecture in algebraic $K$-theory. The map from the colimit over finite subgroups

Let $G$ be a torsion-free group and let $\mathbb{K}$ be a field. The division-ring embedding conjecture. The group algebra $\mathbb{K}[G]$ embeds into a division ring. This is a st…

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, let $G$ be a countable group with a bound on the orders of its finite subgroups, and let $\mathrm{lcm}(G)$ denote the least common m…

Group-ring resolution conjecture. There is a function $f(n)$ such that every $n$-generated ideal in $\mathbb K[G]$ has a free resolution of length $r-1$.

Let $\mathbb{F}$ be a field, let $M$ be a nonsingular matrix over $\mathbb{F}$, and let $x$ be a vector over $\mathbb{F}$. Alon–Jaeger–Tarsi conjecture. For any field $\mathbb{F}$…

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 $G$ be a finitely presented group and let $F$ be a field. Denote by $b^{(2)}_1(G;\mathcal N(G))$ the first $L^2$-Betti number with coefficients in the group von Neumann algebra…

A group $G$ is a Lewin group if, for every field $F$ with a $G$-action, the crossed product $F\ast G$ admits a Hughes-free $F\ast G$-division ring that is its universal division ri…

A Linnell group is a group for which, for every field $F$ and crossed product $F\ast G$, a Linnell division ring $\iota\colon F\ast G\to\mathcal D_{F\ast G}$ exists and is unique u…

Let $G$ be a finite group, let $H\leq G$ be a finite subgroup such that $\operatorname{Cir}_{\mathbb{Q}}^G(H)$ is non-empty, and let $g\in G$ satisfy $H\cap H^g=1$. Let…

Let $G$ be a finite group that is not Dedekind, let $R$ be the coefficient ring, and let $\operatorname{Bic}_R(G)$ be the group generated by all bicyclic units of $RG$. Fix a bicyc…

Surface-by-cyclic group-algebra coherence conjecture. The group algebra $\mathbb{Q}[G]$ is coherent. The source identifies the virtual surface-group case as difficult and open, mot…

Let $G$ be a torsion-free group and let $R$ be a regular ring. Farrell–Jones conjecture for $K_0$. The map induced by the inclusion $R\to RG$ is an isomorphism … This is described…

Let $R$ be a regular ring and let $G$ be a group. The negative K-theory conjecture. If $G$ is torsionfree, then … This is the negative-degree part of the Farrell–Jones program for…

Let $R$ be a field and let $G$ be a group. Kaplansky's idempotent conjecture. If $G$ is torsionfree, then the group ring $RG$ has only the trivial idempotents, namely $0$ and $1$.…

Let $G$ be a group and let $R$ be a regular ring, meaning that $R$ is Noetherian and every $R$-module has a finite-dimensional projective resolution. The regular-ring projective cl…