32 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…

Let $G$ be an amenable group, and let $\lambda_G$ denote its left regular representation. An affine action of $G$ with linear part $\lambda_G$ has an affine action map whose linear…

Spectral approximation conjecture. The normalized eigenvalue-counting functions converge to the spectral density function:

Let $(X,G)$ be a minimal topological dynamical system, where $G$ is an amenable group, and let $\pi_{eq}:X\to X_{eq}$ be the factor map to the maximal equicontinuous factor. Suppos…

Let $(X,G)$ be a minimal topological dynamical system, where $G$ is an amenable group, and suppose that $(X,G)$ admits no essential $K$-IT-tuples. Let $\pi_{eq}:X\to X_{eq}$ be the…

Let $G$ be an amenable group, ${G \curvearrowright X}$ a minimal action, and let $k\geq 2$. Write ${\rm IT}_k(X,G)$ for the set of independence tuples of length $k$, let…

Let $G$ be a countably infinite amenable discrete group acting freely on a finite-dimensional compact metrizable space $X$. Almost finiteness conjecture. The action of $G$ on $X$ i…

Let $G$ be a countably infinite group. Recall that $G$ is richly synergodic if the synergodic actions are dense in $\mathrm{Erg}(G \times G,X,\mu)$ for every standard probability s…

Let $G$ be a second countable locally compact group of type I, meaning that any two irreducible unitary representations of $G$ that are weakly equivalent are unitarily equivalent.…

Real-rank-zero conjecture. Then $G$ is locally finite.

Let $S$ be a finite generating set and let $E$ be a finite set of relators such that … is a finitely presented amenable group. Let the Følner function of $\Gamma$ measure the sizes…

Let $A,B\subseteq\mathbb{R}^n$ be bounded measurable sets, and let $\Gamma$ be a set of isometries of $\mathbb{R}^n$. The sets $A$ and $B$ are $\Gamma$-equidecomposable if they adm…

Positive-relator quotient conjecture.

Let $G$ be a group acting strongly transitively on a totally ordered set $X$ by order-preserving bijections. Amenability conjecture. There exists an amenable group $G$ admitting su…

Friedl–Lück–Tillmann's conjecture. If $G$ is amenable but not virtually $\mathbb{Z}$, then

Let $G\neq \mathbb{Z}$ be an amenable group satisfying the Atiyah Conjecture. Suppose that $G$ is of type $F$ and that $\operatorname{Wh}(G)=0$. The $L^2$-torsion polytope $P(G)$ i…

Let $G$ be a finitely generated group. For a probability measure $\mu$ on $G$, let $A_\mu$ denote the associated subgroup, and let the amenable radical be the largest amenable norm…

Let $G$ be a finitely generated group, and let $\boldsymbol{A_\mu}$ be the subgroup associated with a symmetric, aperiodic probability measure $\mu$ on $G$ whose support generates…

Let $G$ be a finitely generated amenable residually finite group satisfying the ascending chain condition on closed subgroups. Virtual solvability conjecture. The group $G$ is virt…

Let $G$ be an amenable group, let $\mu$ be a probability measure on $G$ with finite second moment and trivial Poisson boundary, and let $u:G\to\mathbb R$ be a Lipschitz $\mu$-harmo…

Gromov's conjecture. The reduced $\ell_p$-cohomology of $G$ vanishes for

Let $\Gamma$ be an amenable group, let $\Sigma$ and $\Sigma'$ be two sofic approximations of $\Gamma$, and let $X$ be a uniformly bounded representation of $\Gamma$. Approximation-…

Let $\Gamma$ be an amenable group and let $Y\subseteq l^{p}(\Gamma)^{\oplus n}$ for some $n\in \mathbb N$. Let $\dim_{l^{p}}^{G}(Y,\Gamma)$ denote the $l^{p}$-dimension defined by…

Gournay's dimension-zero conjecture. One has