Consider a discrete Gromov-hyperbolic group $\Gamma$ (and its Cayley graph $\mathcal{G}$ w.r.t. some generating set). The notion of Gromov-boundary, indicated with $\partial\Gamma$, is naturally associated with the group.

It is well known that two geodesics starting at the same point (assume a vertex) in the graph $\mathcal{G}$ and having the same limit point at infinity (i.e., on the boundary) have to stay at a bounded distance: if $\gamma_1$, $\gamma_2$ are s.t. $\gamma_1(0)=\gamma_2(0)$ and $\gamma_1(\infty)=\gamma_2(\infty)$, then, there exists $K>0$ s.t. $\sup_{t\in [0,\infty)}\mathrm{dist}(\gamma_1(t),\gamma_2(t))<K$, where "$\mathrm{dist}$" denotes the path-distance in the graph.

Let us now specialize to the case where $\Gamma$ has the presentation $\langle a_1,b_1,a_2,b_2\mid [a_1,b_1][a_2,b_2]\rangle$. This is the fundamental group of an orientable, connected and compact surface of genus $2$, for example, a connected sum of two tori.

The Cayley graph of this group is (dual to) a tessellation of the hyperbolic disc by (hyperbolic) octagons, obtained joining at each vertex eight different octagons.

It can be proved that two geodesics $\gamma_1,\gamma_2$ s.t. $\gamma_1(0)=\gamma_2(0)$ and $\gamma_1(n)=\gamma_2(n)$ for some $n\in\mathbb{N}$ satisfy $\sup_{t\in[0,n]}\mathrm{dist}(\gamma_1(t),\gamma_2(t))\leq 4$, meaning in particular that two vertices on the geodesics corresponding to the same parameter $t\in\mathbb{N}$ are at most four edges apart.

My claim is that the same bound holds for the case where $\gamma_1(\infty)=\gamma_2(\infty)$, but $\gamma_1(t)\neq \gamma_2(t)$ for $0<t<\infty$.

Frist of all, I wonder if this claim is actually true. I ask if someone is aware of a counterexample. Then I ask if someone is already in possess of a proof of this claim. I strike that I am referring to a very specific case: I'm interested in the particular group presented above.