87 problems

Let $G$ be a locally compact group. The one-way Cowling conjecture. If $G$ is weakly amenable with $\boldsymbol\Lambda(G)=1$, then it has the Haagerup property. This is the directi…

Automatic continuity conjecture. Every epimorphism from a locally compact Hausdorff group $L$ to $G$ is continuous if and only if $G$ has no non-trivial finite normal subgroups.

Let $T$ be a regular tree and let $G\leqslant\operatorname{Aut}(T)$ be unimodular and transitive on the vertices of $T$. The group C$^*$-algebra of $G$ is CCR if it is liminal, a p…

Let a locally compact group have the approximation property (AP) if there exists a generalized sequence, or net, satisfying the approximation condition for every operator coefficie…

Runde's equality conjecture.

Let $G$ be a locally compact group and let $u$ be a spread out probability measure on $G$. A $u$-boundary is a boundary associated with the measure $u$, and it carries an as…

Massicot–Wagner's conjecture. Even without the definable amenability assumption, a suitable Lie model exists for approximate subgroups.

Multiplier-algebra replacement conjecture. In the corresponding result for $A(G)$, the algebra $C_b(G)$ should be replaced by $MC^*_{\lambda}(G)$.

Operator amenability characterization conjecture. The Fourier–Stieltjes algebra $\mathrm{B}(G)$ is operator amenable if and only if

Weak amenability conjecture for Fourier–Stieltjes algebras. The algebra $\mathrm{B}(G)$ is weakly amenable exactly when $G$ is compact with an abelian connected component of the id…

Let $G=\prod_{i=1}^{m}\mathbf{G}_{i}(k_{i})$, where each $k_i$ is a local field and each $\mathbf{G}_{i}(k_i)$ is the group of $k_i$-points of a Zariski connected simple $k_i$-alge…

Let $G$ be a locally compact group, let $G_d$ denote the same group with the discrete topology, and let $B(G)$ and $B(G_d)$ be their Fourier–Stieltjes algebras. Define the discreti…

Compactness conjecture. Either of these algebras is operator amenable if and only if $G$ is compact.

Let $G$ be a locally compact group equipped with a left-invariant Haar measure, and let $\alpha$ be an automorphism of $G$ that is ergodic with respect to this measure. Halmos's co…

Boidol's conjecture. Every symmetric locally compact group is $*$-regular, and conversely every almost connected, $*$-regular group is symmetric.

Let $G$ be a semisimple Lie group of rank at least $2$, and let a p.m.p. action of $G$ be irreducible, meaning that every factor of $G$ acts ergodically. An action is essentially f…

Let $G$ be a locally compact group. A limit group means a group in the sense of Sela, and $F_2$ denotes the free group on two generators. Dense limit-group conjecture. If $G$ conta…

Let $T$ be a regular tree and let $G\leqslant\operatorname{Aut}(T)$ be a non-amenable automorphism group acting minimally on $T$. Horan–Raum conjecture. The group $G$ is type I if…

Small tripling conjecture. For every $K\ge 1$ and $\varepsilon>0$, this implies

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.…

Let $F$ be a local field of characteristic $0$, let $\mathbf{G}$ be an almost simple, connected and simply connected $F$-algebraic group, and set $G=\mathbf{G}(F)$. In Theorem, whe…

A locally compact group is of type I when its unitary representation theory has the standard type-I decomposition property. A subgroup is cocompact when the quotient by it is compa…

Product conjecture. If $\Gamma$ and $\Gamma'$ are graphically discrete, then

Let $G$ be a locally compact group. For a bounded function space on $G$, write $\mathrm{small}(\mathrm{\mathbf{[NP]}},L^{\infty}(G))$ and…

Computable presentation conjecture. All “natural” groups considered in the field of t.d.l.c. groups have computable presentations, and this can be shown with sufficient effort.