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

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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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  • $\begingroup$ Thank you. Is it possible that there are more than one definitions of divergence around? Bridson-Haefliger (III.H.1.26) prove that if a superlinear divergence function exists, then the group has to be hyperbolic. In your answer you say this is not so. How are these notions of divergence related? $\endgroup$
    – Strichcoder
    Commented Jun 4 at 10:26
  • $\begingroup$ Indeed, the divergence is different in Bridson and Haefliger's book. One is a sup but the other is an inf (of the same quantity). $\endgroup$
    – AGenevois
    Commented Jun 5 at 8:01
  • $\begingroup$ Sorry, I don't see how the two are related. What is the quantity you mention? maybe $\inf \{ length( \text{path} ) : \text{path connects two points outside some ball } \}$? Which definition uses the inf, which uses the sup? Is there an implication between the two? What is the meaning of the condition $e(0) < d(\gamma(R), \gamma'(R))$? Where does the definition in Bridson-Haefliger come from? $\endgroup$
    – Strichcoder
    Commented Oct 7 at 23:25

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