47 problems

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…

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

Peterson–Thom conjecture. There exists a unique maximal amenable von Neumann subalgebra $\mathcal{P}$ of $\mathcal{L}(\mathbb{F}_r)$ such that

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…

Borel Almost Finiteness Conjecture. Every uniformly Borel amenable Følner graph is Borel almost finite.

Amenability–hyperfiniteness conjecture. For countable Borel equivalence relations, Borel amenability is equivalent to hyperfiniteness.

Let $G$ be a semisimple Lie group without compact factors and of rank at least two. A discrete subgroup $\Lambda\le G$ is co-amenable when the Hilbert space $L^2(G/\Lambda)$ admits…

Component-group amenability conjecture. The discrete group

Amenability equivalence conjecture. The following conditions are equivalent:

A II$_1$ factor is nonamenable if it is not amenable, and $L(\mathbb F_2)$ denotes the group II$_1$ factor of the free group on two generators. von Neumann algebraic von Neumann pr…

Inner amenability and decomposable multipliers conjecture. The group $G$ is inner amenable if and only if

Let $G$ be a self-similar group of automorphisms of the infinite rooted $n$-ary tree, and let $V_n(G)$ be the Röver–Nekrashevych group obtained by combining $V_n$ with $G$. Non-inn…

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…

Cancellative free-product conjecture. The semigroup $S$ is a free product of its amenable subsemigroups.

Finite-index free-product conjecture. The semigroup $S$ contains a finite-index subsemigroup which is a free product of its right amenable subsemigroups.

Invariant-means conjecture. The set of left invariant means on $S$ coincides with the set of right invariant means on $S$.

Amenability characterization. $\Gamma$ is amenable if and only if

Let $G$ be a countable group. A representation of $G$ on a Hilbert space is uniformly bounded if there is a constant $K$ such that $\lvert \pi(g)\rvert<K$ for all $g\in G$, and $G$…

Let $T$ be a triangulation, let $x$ be a tangent vector on $T$, and let $G^x$ denote the corresponding graph of discrete tangent vectors. Suppose that $G^x$ is recurrent and that t…

Let $G$ be a finitely generated group, and let $D\Lambda^p_G(n)$ denote its Dirichlet–Poincaré profile for $p\in[1,\infty)$. Grigorchuk–Pansu conjecture for Dirichlet–Poincaré prof…

Let $G(M)$ be a topological group. It is amenable when every $G(M)$-flow admits a $G(M)$-invariant Borel probability measure. Krupiński–Pillay's conjecture. If $G(M)$ is amenable,…

Let $K$ be a compact group, and let $\Lambda_1$ and $\Lambda_2$ be finitely generated subgroups dense in $K$. Lubotzky–Weiss conjecture. If $\Lambda_1$ is amenable and $\Lambda_2$…