Not far not near groups

We use \(NFNN\) as an abbreviation for ‘not far not near’.

Annulus \(A(w, r, R)\) in a metric space \((X, d)\), for \(w \in X, 0 < r < R < \infty \) is defined as the set \[A =\{ x \in X | r \leq d(w, x) \leq R \} \].

Definition

Suppose \(P\) is a finitely generated, one-ended group. Let \(\Gamma(P, S)\) be its Cayley graph with respect to some finite, symmetric generating set \(S\). We say \(P\) is \(M-NFNN\) if there exists an integer \(M\) such that given any two points \(x,y \in \Gamma(P)\) with \(2M \leq r_x \leq r_y\), where \(r_x = d(e,x)\) and \(r_y = d(e,y)\), there exists an arc in \(A(e, \frac{r_x}{3}, 2 r_y) \subset \Gamma(P)\) that connects \(x\) and \(y\).

Question

Characterise NFNN groups.

Remark

Finitely presented groups are \(M-NFNN\) for \(M\) equal to the largest length of a relator.