110 problems

Amenability conjecture. A connected Lie group $G$ has $m(G)<\infty$ if and only if $G$ is amenable.

Peterson–Thom conjecture for hyperbolic groups. If $\mathcal{A}\capmathcal{B}$ is diffuse, then the von Neumann algebra they generate is amenable:

Let $G$ be a group. von Neumann conjecture. The group $G$ is nonamenable if and only if it contains a nonabelian free subgroup. The conjecture was disproved by Olʹshanskiĭ in 1980;…

A group is non-amenable if it does not admit an invariant mean, and a non-abelian free subgroup is a free group on at least two generators. Von Neumann–Day conjecture. A group is n…

Runde's equality conjecture.

Let $G$ be a graph equipped with a nonunimodular transitive automorphism group, and let $p_h$ and $p_u$ denote the Hausdorff and uniqueness thresholds, respectively. Suppose that t…

Let an automaton group have degree $2$, meaning that every element $g$ has activity $a_n(g)=O(n^2)$, and let the associated degree-$2$ mother groups be the specific universal autom…

Let $G$ be a group definable over $M$. A definable action of $G(M)$ on a compact space is an action for which each orbit map is definable. The group is weakly definably amenable wh…

Let $L(\mathbb{F}_n)$ be a free group factor and let $B\subset L(\mathbb{F}_n)$ be a diffuse amenable subalgebra. A maximal amenable extension of $B$ is an amenable subalgebra of…

The Tarski alternative. Exactly one of the following is true:

Grigorchuk–Nagnibeda conjecture. The group $G$ is amenable if and only if $\rho=\rho_r$.

Complete convergence conjecture. Complete convergence should hold more generally when $a=a^\dagger$ and $\Lambda$ is amenable, but not in general on nonamenable lattices. Complete…

Let $\textbf{AM}$ be the class of amenable groups, and let $\textbf{NFree}$ be the class of groups that do not contain the free group $F_2$ as a subgroup. Von Neumann's conjecture.…

Let $\mathcal{A}$ be a Banach algebra. It is super-amenable (or contractable) if every bounded derivation $d:\mathcal{A}\rightarrow E$ into every Banach $\mathcal{A}$-bimodule $E$…

Converse Liouville property conjecture. Any amenable measured groupoid is Liouville.

Let $G$ be an infinite, finitely generated group. Let $M$ be a Banach space with norm $||\ ||_M$ that is also a left $\C G$-module, and suppose that $G$ acts on $M$ by continuous…

Converse conjecture for $B(G)$. If $B(G)$ has a normal, virtual diagonal, then $G$ has an abelian subgroup of finite index; moreover, the same converse should hold with the existen…

Let $G$ be a locally compact group, let $A(G)$ be its Fourier algebra, and let $OA(G)$ denote the operator space structure over $A(G)$ introduced by the author of the cited work. R…

Let $F$ denote R. Thompson's group, a finitely presented group known not to contain non-abelian free subgroups. Geoghegan's conjecture. The group $F$ is not amenable. A considerabl…

Let $L_p$ be the usual $L_p$ space, with $1\leq p<\infty$, and equip its isometry group $\operatorname{Iso}(L_p)$ with the strong operator topology. Extreme amenability conjecture.…

Let $\Gamma$ be a group, let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$, and write $\Gamma^\mathcal{U}$ for its ultrapower. A group is uniformly exact when the action of…

Let $G$ be a group. A set $R\subset G$ is a set of measurable recurrence if, for every measure-preserving $G$-system $(X,\mathcal{X},\mu,(T_g))$ and every $A\in\mathcal{X}$ with…

A supramenable group is a group $G$ such that every $G$-set $X$ supports an invariant finitely additive measure normalized on any given non-empty subset $A\subseteq X$. The fixed-p…

A group $G$ is framenable if it has the framenability property defined in the paper, and a group has property (T) in the sense of Kazhdan. Framenability versus property (T) conject…

Let $G$ be a graph whose automorphism group contains a nonunimodular subgroup acting quasi-transitively. Let $p_c$, $p_h$, and $p_u$ denote respectively the critical, heavy-cluster…