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Let $G$ be a finitely generated hyperbolic group with the word metric; fix a symmetric generating set $S$ and let $\mathcal{G}$ be the Cayley graph of $G$ w.r.t. $S$. Define the cone of an element $x\in G$ as the set of all $z\in G$ such that there exists a geodesic in $\mathcal{G}$ starting from (the vertex correspondent to) $e$, passing through $x$ and terminating in $z$. Denote the cone type of $x$ with $C(x)$.

Pick two elements $x_1,x_2\in G$. My question: can we write the intersection $C(x_1)\cap C(x_2)$ as a finite union of cones?

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  • $\begingroup$ The answer should be "yes" because every geodesic of a hyperbolic group is Morse. $\endgroup$
    – user6976
    Commented Oct 26, 2017 at 11:15
  • $\begingroup$ Maybe I can try to deal with a very simple example to get more insight: I am particularly interested in surface groups. Let's fix the genus $g \geq 2$ and let's consider the surface group $\Gamma_g$. Then, given a finite geodesic in its Cayley graph, any other geodesic joining the same endpoints is in a $2g$-neighborhood of the first, right? $\endgroup$
    – EM90
    Commented Oct 26, 2017 at 12:12
  • $\begingroup$ Certainly - in some neighborhood of bounded diameter. I am not sure about $2g$, but it is quite possible (may be even less). For surface groups it was proved essentially by Dehn. $\endgroup$
    – user6976
    Commented Oct 26, 2017 at 15:03

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