35 problems

Bader–Muchnik irreducibility conjecture. For every locally compact group $G$ and every spread-out probability measure … -boundary of $G$ is irreducible.

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 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. 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…

Let $G$ be a locally compact group. The Haagerup property equivalence conjecture. $G$ has the Haagerup property if and only if $G$ is weakly amenable with $\boldsymbol\Lambda(G)=1$…

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…

Let $T$ be a thick semi-regular tree, and let $G$ be a closed subgroup of $\operatorname{Aut}(T)$ acting transitively on the boundary $\partial T$. A locally compact group $G$ is C…

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 $G$ be a second countable locally compact group. Furstenberg-boundary simplicity conjecture. If some point of the Furstenberg boundary of $G$ has a trivial stabilizer, then … T…

Let $G$ be a second countable locally compact group. Type I group conjecture. If $G$ is typeI, then $G$ admits a cocompact amenable subgroup. This proposes a structural consequence…

Scaling-group conjecture. The scaling group of $G$ is a subgroup, possibly trivial, of

Wavelet completeness conjecture. Characterize the locally compact groups $G$ satisfying

Fix $k$ and let $\tau$ be an irreducible representation of $\mathrm{GL}(k,\mathbb F)$. Consider the triples $(z,k,\tau)$ that correspond to unitary representations of…

Let $\rho$ be an irreducible unitary representation of $\overline{\mathrm{GL}}$, and let $z(\rho)$ denote its spherical character. The preceding construction gives…

Figà-Talamanca–Picardello conjecture. $A_r(G)$ is dense in $B_0(G)$, equivalently

Harmonic-function dimension conjecture. For every $k\geq 0$,

Let $G$ be a nilpotent locally compact $p$-group, and let $\mathrm{rank}_p(G)$ denote its $p$-rank. The finite-rank conjecture. If … then $G\in\mathfrak E_{<\infty}$, meaning that…

Spectrally reasonable measures conjecture.

For a locally compact group $G$ and $1\leq p\leq\infty$, let $C^*_{L^p}(G)$ and $C^*_{L^{p+}}(G)$ denote the corresponding exotic group C-algebras, and let $B_{L^p}(G)$ be the asso…

Let $G$ be a Kunze–Stein group, and let $\pi:G\to B(H_\pi)$ be a unitary representation admitting a cyclic vector $\xi\in H_\pi$. Write…

Let $\mathfrak{A}$ be an order in a finite-dimensional semisimple algebra, and let $M$ be a discrete object of $\mathsf{LCA}_{\mathfrak{A}}$. Projective resolutions of $M$ in…

Adelic-block conjecture. Every adelic block $G\in\mathsf{LCA}_{F}$ is both an injective and projective object.

Nilpotent-by-finite characterization. The group $G$ has a finite-index nilpotent subgroup if and only if $\dim HF_k(G,\mu)<\infty$ and there exists a finite-index subgroup $H$ of…