Let $g \geq 2$ and $S_g$ be a closed, connected and orientable genus $g$ surface. Fix a covering map $\mathbb{H}^2 \to S_g$. Identify $\pi_1(S_g)$ with the deck transformation group. For any $\gamma \in \pi_1(S_g)$ this yields an isometry of hyperbolic type on $\mathbb{H}^2$ with a geodesic axis on which it acts by translation. For $\gamma_1, \gamma_2 \in \pi_1(S_g)$, we say that $\gamma_1, \gamma_2$ are linked at infinity if the endpoints of one axis separates the endpoints of the other axis and vice-versa.
Let $\alpha, \beta, \gamma \in \pi_1(S_g)$ be such that their axes are pairwise disjoint, the axes for $\alpha$ and $\beta$ are on the same side of the axis for $\gamma$ and they are pairwise not linked at infinity. How does one follow that there exists $\delta \in \pi_1(S_g)$ that is linked at infinity with $\alpha, \beta$ but not with $\gamma$?
It is clear to me that one can easily draw a geodesic in $\mathbb{H}^2$ intersecting the axes of $\alpha$ and $\beta$ but not $\gamma$, but why can one draw such a geodesic that is the axis of an element in $\pi_1(S_g)$?
This is one direction of the "iff" statement of the proof of Corollary 8.6 in Farb and Margalit's A Primer on Mapping Class Groups.
Here's a definition and a theorem.
Definition: A 2 point subset $\{\xi,\eta\} \subset \partial \mathbb{H}^2$ is called an axis endpoint pair if there exists an element of $\pi_1(S_g)$ having axis $L$ such that $\partial L = \{\xi,\eta\}$.
Theorem: The set of axis endpoint (ordered) pairs is dense in $\partial \mathbb{H}^2 \times \partial\mathbb{H}^2$.
So the geodesic you were able to draw, the one that intersects the axis of $\alpha$ and $\beta$ but not of $\gamma$, may be approximated by another geodesic having the same intersection pattern and which is the axis of an element of $\pi_1(S_g)$.
Here's a sketch of a proof of the theorem. Let $D$ be the diameter of $S_g$ in its hyperbolic metric.
Consider any 2 point subset $\{\xi,\eta\}$. Let $L \subset \mathbb{H}^2$ be the line such that $\partial L = \{\xi,\eta\}$. Choose a finite subsegment $\tilde\ell \subset L$ whose two (finite) endpoints $x,y$ are very close to $\xi,\eta$ (resp.) in $\mathbb{H}^2 \cup \partial\mathbb{H}^2$, so close that $\text{Length}(L) > D$. Project $\tilde\ell$ down to $S_g$, obtaining a finite geodesic segment $\ell$. Choose $m$ to be any finite geodesic segment with endpoints $y,x$ of length $\le D$. The closed path $\ell * m$ is homotopically nontrivial. By choosing the correct lift of $m$ one obtains a concatenated segment $\tilde\ell * \tilde m$, and an element $\gamma$ taking the initial point of $\tilde\ell*\tilde m$ to the terminal point; the conjugacy class of this element $\gamma$ is represented in $\pi_1(S_g)$ by the closed curve $\ell * m$. This path $\tilde\ell*\tilde m$ can be concatenated bi-infinitely with its translates under powers of $\gamma$ to form a piecewise geodesic axis for $\gamma$ that I'll denote $L'(\gamma)$. Furthermore, this piecewise geodesic line $L'(\gamma)$ converges at its two ends to the two endpoints $\{\xi',\eta'\} \subset \partial\mathbb{H}^2$ of the true geodesic axis $L(\gamma)$ of $\gamma$.
Now one does some estimates to verify that $\xi',\eta'$ are close to the endpoints $\xi,\eta$ of the original line $L$, which finishes the proof of the theorem.
The underlying technicalities of these estimates show that for any neighborhoods $U_\xi,U_\eta \subset \mathbb{H}^2 \cup \partial\mathbb{H}^2$ of $\xi,\eta$ respectively, there exist (smaller) neighborhoods $V_\xi,V_\eta$ such that if $x \in V_\xi$ and $y \in V_\eta$ then $\xi' \in U_\xi$ and $\eta' \in U_\eta$.
The underlying mathematical concept behind this argument is the concept of a "quasigeodesic". The estimates one proves are driven by the need to show that the piecewise geodesic line $L'(\gamma)$ is a $K,C$-quasigeodesic with a very good multiplicative distortion constant $K$ (arbitrarily close to $1$) and with a controlled additive distortion constant $C$ (uniformly bounded away from $\infty)$.