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 $\lim\inf_{n\to\infty}\frac{e(n)}n = \infty$.


1 Answer 1


There exist many non-hyperbolic groups with superlinear divergence. For instance:

  • One-ended relatively hyperbolic groups have exponential divergence. (See Sisto's article On metric relative hyperbolicity.)
  • One-ended acylindrically hyperbolic groups have superlinear divergence.
  • For every integer $d \geq 1$, there exists a right-angled Coxeter group whose divergence is polynomial of degree $d$. (See Dani and Thomas' article Divergence in right-angled Coxeter groups.)
  • There exist also groups with exotic divergence functions, coming for instance from small cancellation groups or lacunary hyperbolic groups.

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