Weakly $G$-slim complexes and the non-positive immersion property for generalized Wirtinger presentations

We investigate weakly $G$-slim complexes, a more flexible variant of Helfer and Wise's slim complexes, which can be defined on any regular $G$-covering. We prove that if a $2$-complex $X$ associated to a group presentation is weakly $H$-slim for some regular $H$-covering, and $\pi_1X$ is left-orderable, then $X$ is slim and, in particular, it has non-positive immersions. We apply this result to study conditions on generalized Wirtinger presentations that guarantee the non-positive immersion property for their associated $2$-complexes. This class of presentations includes classical Wirtinger presentations of (perhaps high-dimensional) knots in spheres, some classes of Adian presentations and LOTs presentations. On the one hand, our results provide a wide variety of examples of presentation complexes with the non-positive immersion property via a condition that can be checked with a simple algorithm. On the other hand, as an application of these methods, we derive an extension of a result by Wise on Adian presentations and prove a stronger formulation of a result of Howie on LOT presentations.