14.98 (1999)

We say that a metric space is a 2-end one (a narrow one), if it is quasiisometric to the real line $\mathbb{R}$ (respectively, to a subset of $\mathbb{R}$). All other spaces are said to be wide. Suppose that the Caley graph $\Gamma = \Gamma(G, A)$ of a group $G$ with a finite set of generators $A$ in the natural metric contains a 2-end subset, and suppose that there is $\varepsilon > 0$ such that the complement in $\Gamma$ to the $\varepsilon$-neighbourhood of any connected 2-end subset contains exactly two wide connected components. Is it true that the group $G$ in the word metric is quasiisometric to the Euclidean or hyperbolic plane?