13.61 (1995)

We call a metric space narrow if it is quasiisometric to a subset of the real line, and wide otherwise. Let $G$ be a group with the finite set of generators $A$, and let $\Gamma = \Gamma(G, A)$ be the Cayley graph of $G$ with respect to the natural metric. Suppose that, after deleting any narrow subset $L$ from $\Gamma$, at most two connected components of the graph $\Gamma \setminus L$ can be wide, and there exists at least one such a subset $L$ yielding exactly two wide components in $\Gamma \setminus L$. Is it true that $\Gamma$ is quasiisometric to an Euclidean or a hyperbolic plane?