2 problems

Bodart–Tashiro conjecture. The group $G$ is virtually Abelian if and only if it admits a Cayley metric whose metric-functional boundary is countable.

Weak gap conjecture. If $G$ has growth $\preceq r\mapsto\exp(r^\alpha)$ for some $\alpha<\tfrac12$, then there exists a Banach metric $d$ on $G$ such that $\partial(G,d)$ contains…