Order-detection and non-left-orderable surgeries on links

Beginning with a $3$-manifold $M$ having a single torus boundary component, there are several computational techniques in the literature that use a presentation of the fundamental group of $M$ to produce infinite families of Dehn fillings of $M$ whose fundamental groups are non-left-orderable. In this manuscript, we show how to use order-detection of slopes to generalise these techniques to manifolds with multiple torus boundary components, and to produce results that are sharper than what can be achieved with traditional techniques alone. As a demonstration, we produce an infinite family of hyperbolic links where many of the manifolds arising from Dehn filling have non-left-orderable fundamental groups. The family includes the Whitehead link, and in that case we produce a collection of non-left-orderable Dehn fillings that precisely matches the prediction of the L-space conjecture.