81 problems

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

Weiss's conjecture. Every orbit equivalence relation induced by a Borel action of a countable amenable group is hyperfinite.

Let $C$ be a compact group containing finitely generated dense subgroups $G$ and $H$, where $G$ is amenable and $H$ has property $(\operatorname{T})$. Lubotzky–Weiss conjecture. Th…

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…

A probability measure on a group is non-degenerate if its support is not contained in a proper subgroup. A probability measure is Liouville when every bounded harmonic function for…

Let $G$ be an amenable group acting minimally and topologically freely on a compact metric space $X$. The action has mean dimension zero when its mean dimension, in the sense of Gr…

A group is amenable if it has an invariant mean, and it satisfies Lück's dimension-flatness if, for every integer $j\geq 1$ and every $\mathbb Z\Gamma$-module $\mathcal M$, one has…

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-…

Gournay's dimension-zero conjecture. One has

Let $\Gamma$ be a finitely generated amenable group, and let $p$ be a real number greater than one. Its first reduced $\ell^2$-cohomology is known to be zero. Gromov's conjecture.…

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…

Gromov's conjecture. Every finitely generated amenable group has no first reduced $\ell^p$-cohomology.

Amenable-group equality conjecture. For every amenable group $G$, is it true that

Let $G$ be an amenable finitely generated group. Say that $G$ can be approximated by elementary amenable groups when it is a limit of such groups in the space of marked groups. Ste…

Bounded-class elementary amenable–$F$ alternative conjecture. Every subgroup of $PL_o(I)$ is either elementary amenable of some class $lpha$ with

Elementary amenable–$F$ alternative conjecture. Every subgroup of $PL_o(I)$ is either elementary amenable or it contains a subgroup that is isomorphic to $F$.

Let $G$ be a locally compact group, and let $A(G)$ denote its Fourier algebra. Amenability equivalence conjecture. The group $G$ is amenable if and only if the Banach algebra…

Amenable-group complete contraction conjecture. The identity map from $A(G)$ to $OA(G)$ is a complete contraction for each amenable $G$.

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

Let $M$ be a compact manifold, let … be a noncompact amenable Galois cover, and let $\{D_k\}_{k=1}^{\infty}$ be a regular exhaustion of $\widehat{M}$. Write $b^j(D_k,\partial D_k)$…

Exponential conjugacy-growth conjecture. The conjugacy growth of $G$ is exponential.

Let $G$ be a countable amenable group. An IP$^{*}$-set in $G$ is a set meeting the image of every increasing IP sequence, where an increasing IP sequence is a map…

Let $G$ be a finitely generated group, and let $S$ be a finite generating set. Write $\mathrm{cp}_S(G)$ for the limiting commuting probability associated with the word-ball exhaust…

Almost-finiteness conjecture. Every free action of an amenable group on the Cantor set is almost finite.

Let $G$ be a discrete group. Its amenable radical is the largest normal amenable subgroup of $G$, and $G$ is $C^*$-simple when its reduced $C^*$-algebra has no non-trivial closed t…