Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.

We note that in $\mathbb{R}^2$ there is no divergence function at all: take rays with arbitrarily small angle $\varphi$, so that paths outside $B(r+R)$ have approximately size $\varphi\cdot(r+R)$. There is no function $e(r)$ that is smaller than $\varphi\cdot (r+R)$ for all $\varphi$.

There are metric spaces with linear divergence functions, for example consider the graph consisting of two rays where the $n$th vertices are connected by a segment of length $n$.

Q: Are there non-hyperbolic groups whose Cayley-graphs admit a divergence function?

The following definition is from the community wiki post, see also Definition III.H.1.24 and Proposition III.H.1.26 in *Metric spaces of non-positive curvature* by Bridson and Haefliger.

**The super-linear divergence of geodesics condition.** Let $X$ be a geodesic metric space. A map $e\colon \mathbb{N} \to \mathbb{R}$ is a *divergence function* for $X$ if for all $R$, $r$ in $\mathbb{N}$, all $x \in X$ and all geodesics $\gamma\colon [0,a_1]\to X$ and $\gamma'\colon [0,a_2] \to X$ with $\gamma(0) = \gamma'(0) = x$ such that $R + r \le \min\{a_1,a_2\}$ and $d(\gamma(R),\gamma'(R)) > e(0)$, then we have that any path connecting $\gamma(R+r)$ to $\gamma'(R+r)$ outside the ball $B(x,R+r)$ has length at least $e(r)$.

A divergence function $e$ is super-linear if $\lim\inf_{n\to\infty}\frac{e(n)}n = \infty$.