# Divergence functions in hyperbolic groups

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

• 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.)