125 problems

Let $G$ be a torsion-free group, let $\mathbb{F}$ be a field, and let $\mathbb{F}[G]$ be the group ring. The obvious units in $\mathbb{F}[G]$ are the elements $\lambda g$ with…

For a group $G$, let $\mathbb{Z}G$ be its integral group ring, let $K_{0}(\mathbb{Z}G)$ be its Grothendieck group of finitely generated projective $\mathbb{Z}G$-modules, and let ……

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 $\mathbb{F}$…

Hsiang's conjecture.

Let $G$ be a group, let $r \colon K_0\mathbb{Z}G\rightarrow HH_0(\mathbb{Z}G)$ be the Hattori–Stallings trace map, and write the image of a finitely generated projective module $P$…

Let $G$ be a torsion-free countable group, let $k$ be a subfield of $\mathbb{C}$, and let $\D(k[G])$ be the Linnell ring, defined as the division closure of $k[G]$ in the ring of u…

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 $R[G]$ denote the group ring of a group $G$ over a ring $R$. Baer's conjecture. The group ring $R[G]$ is Noetherian if and only if $G$ is virtually polycyclic. The source state…

Let $G$ be a group, let $\mathbb C G$ be its complex group algebra, and let $\ell^2(G)$ be the Hilbert-space completion of $\mathbb C G$ in the 2-norm. Linnell's analytic zero divi…

Let $G$ be a torsion-free group, and let ${\mathbb Z}(G)$ denote its integral group ring. Each element of ${\mathbb Z}(G)$ acts by left convolution on $\ell^{2}(G)$. Atiyah conject…

Let $k$ be a regular ring, let $G$ be a torsionfree group, let $\mathrm{B}G$ denote its classifying space, and let $\mathbf{K}(k)$ denote the algebraic $K$-theory spectrum of $k$.…

Rationalized Farrell-Jones conjecture. For every group $G$ and every $n\in\mathbb{Z}$, this rationalized Farrell-Jones assembly map is an isomorphism.

Bass's Whitehead-group conjecture. The group $\mathit{Wh}(G)$ vanishes.

Let $G$ be a group, let $R$ be a ring, and consider the assembly map … Here $BG$ is the classifying space of $G$, $K(R)$ is the algebraic $K$-theory spectrum of $R$, and $R[G]$ is…

Let $p\colon\Gamma\to G$ be a surjective group homomorphism, and let $p^*\operatorname{vcyc}$ be the family of subgroups of $\Gamma$ whose images under $p$ are virtually cyclic. Le…

Matrix-kernel approximation conjecture.

Let $G$ be a torsionfree group, and let $C^*_r(G)$ denote its reduced group $C^*$-algebra. An idempotent is an element $e$ satisfying $e^2=e$. Kadison's conjecture. Every idempoten…

Throughout, let $p$ be a prime, let $C_p={\bold Z}/p{\bold Z}$ with generator $\alpha$, and let $R={\bold Z}C_p$. For an $R$-module $M$, let $\pi_M:{\bold Z}M\to M$ be the canonica…

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…

Let $H$ be a torsion-free group, and let $\mathbb{Z}H$ denote its integral group ring. Write $\pm H$ for the subgroup generated by $H$ and the two-element group generated by $-1$.…

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 $M2)$…

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…

Zero-divisor conjecture. If $G$ is torsion-free, then $\mathbb{Q}[G]$ has no non-trivial zero divisors.

Zero-Divisor Conjecture. The ring $\mathbb C\Gamma$ contains no zero-divisors.

Let $\Gamma$ be a torsion-free group. Consider the maps … and … defined by the corresponding $L^2$-signature and ordinary signature constructions. Torsion-free group-ring signature…