1
$\begingroup$

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

$\endgroup$

1 Answer 1

2
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.