# 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.

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