25 problems

Let $\mathbb{F}$ be a skew field. Let $G$ be a torsion-free residually finite group of type $\mathsf{FP}_{n+1}(\mathbb F)$ such that the Hughes-free division ring…

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 $g>0$. A simple tiling $\mathcal T$ of a genus-$g$ surface has an associated incidence theorem over a division ring ${\mathrm D}$, and $n$ denotes the dimension of the ambient…

Let $\mathcal T$ be a simple tiling of a surface of positive genus, meaning that its underlying graph is $(n+1)$-degenerate, and let ${\mathrm D}$ be a division ring. The incidence…

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…

Additive-map factorization conjecture. There exists a fixed element $q\in D$ such that

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…

Strict containment conjecture. The class of weakly locally finite division rings strictly contains the class of Stewart's division rings.

Let $n\in\mathbb{N}$ with $n>1$, and let $\varphi:F\to M_n(G)$ be a ring embedding, where $G$ and $F$ are division rings. Suppose that $a$ and $b$ are positive integers satisfying…

Let $D$ be a non-commutative division ring algebraic over its center. A division subring is centrally finite if it has finite dimension over its center. Mahdavi-Hezavehi's weak Kur…

Let $D$ be a division ring with center $F$. A subfield $K$ of $D$ is maximal if it is maximal among subfields of $D$. Maximal-subfields algebraicity conjecture. If every maximal su…

Let $D$ be a division ring and let $D^*$ be its multiplicative group. A subgroup is non-central if it is not contained in the center of $D$. Weak Herstein–Scott conjecture. Every n…

Let $L/K$ be an extension of subfields of $\mathbb{C}$, and let $G$ be a group satisfying the strong Atiyah conjecture. Let $\mathcal D_{K[G]}$ and $\mathcal D_{L[G]}$ be, respecti…

Solvable maximal subgroup conjecture. If $n\geq 2$, then $G$ contains no solvable maximal subgroups.

Subnormal subgroup algebraicity conjecture. If $N$ is non-central, then $N$ is left algebraic (respectively, right algebraic) over $K$ if and only if $D$ is left algebraic (respect…

Let $D$ be a division ring with center $F$, and let $\operatorname{GL}_n(D)$ be the general linear group of degree $n$ over $D$. If $n\geq 2$, assume that $D$ is not a locally fini…

Makar-Limanov's involutional conjecture. If $D$ is infinite dimensional over $k$ and finitely generated as a division $k$-algebra, then there exist two symmetric elements in $D$ wh…

Let $D$ be a division ring with center $k$. Let a free group algebra over $k$ mean the group algebra over $k$ of a free group embedded in $D$. Free group algebra conjecture. If $D$…

Let $D$ be a division ring, and write $D^\times$ for its multiplicative group. A group is noncyclic free if it is a free group that is not cyclic. Lichtman's conjecture. The group…

Let $A$ be a noetherian, connected graded domain over ${\Bbbk}$ with $\operatorname{GKdim} A=3$. Its graded quotient ring has the form $Q_{gr}(A)\cong D[t,t^{-1};\alpha]$, where…

Let $D$ be an infinite division ring, let $n$ be a natural number, and let $M$ be a maximal subgroup of $\mathrm{GL}_n(D)$. Non-polycyclic-by-finite maximal subgroup theorem. If…

Let $D$ be an infinite division ring, let $n$ be a natural number, let $N$ be a subnormal subgroup of $\mathrm{GL}_n(D)$, and let $M$ be a nilpotent maximal subgroup of $N$. Nilpot…

Bien's conjecture. Every normal subgroup of $D^*$ that is radical over $K$ is commutative.