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I am reading this paper on boundaries of hyperbolic groups. In this paper, a geodesic metric space $(X,d)$ is considered. A sequence of points $(x_n)_{n \geq 1}$ converges to infinity if $$\lim \inf_{i,j \to \infty} (x_i, x_j)_x = \infty$$ where $(x_i, x_j)_x$ is the Gromov product with respect to the point $x$, that is $$(x_i,x_j)_x = \frac{1}{2}(d(x_i,x) + d(x_j,x) - d(x_i,x_j))$$

The paper then states that for a geodesic ray $\gamma: [0, + \infty) \to X$ the sequence $\gamma(n)$ converges to infinity, which I tried to prove, but which I couldn't prove.

Any help would be appreciated!

EDIT: Using @Seirios' hint, I came up with the following reasoning: We first compute $(\gamma(i),\gamma(j))_{\gamma(0)}$. Assuming $i \leq j$, we find that $$(\gamma(i),\gamma(j))_{\gamma(0)} = \frac{1}{2}(d(\gamma(i), \gamma(0)) + d(\gamma(j), \gamma(0)) - d(\gamma(i), \gamma(j)))$$ and since $\gamma$ is a geodesic ray, we have that $d(\gamma(t), \gamma(t')) = |t - t'|$, hence the above equation simplifies to $$(\gamma(i),\gamma(j))_{\gamma(0)} = \frac{1}{2}(i + j - j + i) = i.$$

By Seirios' second hint, we have that for any $x \in X$ $$(\gamma(i),\gamma(j))_{\gamma(0)} - d(\gamma(0),x) \leq (\gamma(i),\gamma(j))_x$$ where we can assume $d(\gamma(0),x)$ to be finite (since the limit notion does not depend on $x$. Taking limes inferior on both sides shows that the left hand side goes to $\infty$ and hence the right hand side must too.

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  • $\begingroup$ You give no indication of what problem you are having proving it. What have you tried? Do you have a picture in mind of what the Gromov product is? $\endgroup$ – Paul Plummer Mar 10 '18 at 17:57
  • $\begingroup$ @PaulPlummer I only know the definition of gromov product and I am not really sure how to visualize it. $\endgroup$ – Student Mar 10 '18 at 18:35
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Hint 1: First compute $\left( \gamma(i),\gamma(j) \right)_{\gamma(0)}$, and next compare $\left( \gamma(i),\gamma(j) \right)_{\gamma(0)}$ and $\left( \gamma(i),\gamma(j) \right)_{x}$ for an arbitrary $x \in X$.

Hint 2: Notice that, for every $x,y,u,v \in X$, one has $(x,y)_u \leq (x,y)_v + d(u,v)$.

The intuition behind the Gromov product is that $(y,z)_x$ quantifies the amount of time two geodesics $[x,y]$ and $[x,z]$ fellow travel. To convince yourself:

  • compute the Gromov product of three vertices in a tree;
  • in a $\delta$-hyperbolic space $X$, given three points $x,y,z \in X$ and two geodesics $[x,y]$ and $[x,z]$, compute the distance between $y_t$ and $z_t$ for every $t \leq (y,z)_x$, where $y_t$ and $z_t$ denote the points of $[x,y]$ and $[x,z]$ respectively at distance $t$ from $x$.
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  • $\begingroup$ Thanks for your answer. I did the first part, but don't see what the result should be from comparing both Gromov distances w.r.t. $\gamma(0)$ and $x$... For the intuition part, a tree is $0$-hyperbolic, so I assume both points are distinct cases? $\endgroup$ – Student Mar 11 '18 at 18:04
  • $\begingroup$ I added a second hint. For the tree, just draw a picture, and identify the Gromov product with the length of some geodesic. $\endgroup$ – Seirios Mar 12 '18 at 7:55
  • $\begingroup$ For a tree, I think I was able to prove that the initial segment exactly has the Gromov product. For the second one, in a general $\delta$-hyperbolic space, I tried to show they are on a distance $2\delta$ from one another by using the triangle definition, but I did not succeed. For the limit, I have edited my answer, could you check it please? Thank you! $\endgroup$ – Student Mar 12 '18 at 10:24
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    $\begingroup$ You get the correct answer for the tree, and your solution to your initial problem is correct as well. A good thing to do for your last problem would be understand the equivalences between all the definitions of hyperbolicity given in the book Sur les groupes hyperboliques d'après M. Gromov, partially translated in English here: perso.ens-lyon.fr/ghys/articles/groupeshyperboliques-english.pdf. $\endgroup$ – Seirios Mar 13 '18 at 8:14

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