Every compact 3-Sasakian 7-manifold $M$ admits a canonical 2-parameter family of co-closed $\text{ G}_2$-structures $\varphi_{ a,b}$ for $a,b > 0$, as well as a foliation by $\varphi_{ a,b}$-associative 3-folds whose leaf space $X$ is a positive quaternion-K\"{ a}hler 4-orbifold. We prove that associative 3-folds in $(M,\varphi_{ a,b})$ that are ruled by a certain type of geodesic are in correspondence with pseudo-holomorphic curves in the almost-complex 8-manifold $Z \times S^2$, where $Z$ is the twistor space of $X$ equipped with its strict nearly-K\"{ a}hler structure. As an application, we construct infinitely many topological types of non-trivial, compact associative 3-folds in the squashed 7-spheres $(S^7, \varphi_{ a,b})$ and squashed exceptional Aloff-Wallach spaces $(N_{ 1,1}, \varphi_{ a,b})$. Topologically, our examples are circle bundles over a genus $g$ surface, for any $g \geq 0$.
Categories: math.DG