7
$\begingroup$

Let $\delta(G)$ be the smallest Gromov-hyperbolicity constant of a finite group $G$ (e.g., for any generating set of $G$, all triangles in the corresponding Cayley graph are $\delta(G)$-thin). Let $S_n$ be the symmetric group on $n$ letters. Does this limit exist?

$$\lim_{n \to \infty} \delta(S_n)$$

(If not, is there anything known about the asymptotic behavior?) More generally, are there non-trivial sequences of finite groups that are asymptotically hyperbolic in this sense?

Improve this question
$\endgroup$
8
  • $\begingroup$ There is a constant $\delta$ such that,for any hyperbolic group $G$, finite or infinite, there is a generating set $X$ for $G$ such that all triangles in the Cayley graph with respect to $X$ are $\delta$-thin. This is true with $\delta=14$, for example, but that is unlikely to be optimal. $\endgroup$
    – Derek Holt
    Jul 28, 2017 at 19:23
  • 4
    $\begingroup$ For the generating subset $S=G$, $\delta=1$. So $\delta(F)\le 1$ for every finite group $F$ (and $=1$ if $|F|\ge 3$). So the question is trivial. Maybe one should work with a bound on $|S|$ and let $G$ vary. $\endgroup$
    – YCor
    Jul 28, 2017 at 19:29
  • 3
    $\begingroup$ I am sure that $\delta(S_n)$ tends to infinity with the generators $(1,2),(1,2,3,\ldots,n)$. Even the bigon thinness constant does! For $n$ even, you could take the bigon with sides labelled $g^{n/2}$ and $g^{-n/2}$, where $g$ is the second generator. $\endgroup$
    – Derek Holt
    Jul 28, 2017 at 19:45
  • 1
    $\begingroup$ You may be interested in this paper of Berestycki statslab.cam.ac.uk/~beresty/Articles/Geometry-final.pdf $\endgroup$
    – tmh
    Jul 28, 2017 at 21:42
  • 3
    $\begingroup$ A less trivial question would ask for the limit $\lim \frac {\delta(S_n)}{n}.$ $\endgroup$
    – user6976
    Jul 29, 2017 at 1:47

0

Reset to default

Your Answer

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