7 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…

Cornulier–Stalder–Valette's conjecture. The Haagerup property for $G$ and $H$ should imply the Haagerup property for every permutational wreath product $G\wr_X H$.

Let $\mathcal{G}$ be a Hausdorff étale groupoid. Weak amenability means that $\Lambda_{\text{cb}}(\mathcal{G})$ is finite, and the condition $\Lambda_{\text{cb}}(\mathcal{G})=1$ is…

Converse Haagerup-property conjecture. If $B\wr^A_X H$ has the Haagerup Property, then for every point stabilizer $L$ of $X$, the relative profinite completion…

Regular equivalence-class conjecture. There exists $2^{\aleph_0}$ regular equivalence classes of Haagerup monsters.

Let $X$ be a compact ultrametric space, and let $LS(X)$ denote the locally compact group of local similarities on $X$. Haagerup property conjecture. The group $LS(X)$ has the Haage…

Nica's conjecture. For discrete countable groups, the converse implication in the first assertion and the direct implication in the second assertion hold: if every continuous isome…