Since the pioneering work of Thurston on hyperbolic geometry, and of Neumann-Zagier on volume and triangulations, it has been believed that every cusped hyperbolic \( 3 \)-manifold should admit a decomposition into a union of positively oriented ideal tetrahedra. Somewhat shockingly, the question of whether such a geometric triangulation exists is still open today. Luo, Schleimer, and Tillmann proved that geometric ideal triangulations of this sort exist in some finite cover of every cusped \( 3 \)-manifold. We extend their result by showing that every cusped hyperbolic \( 3 \)-manifold has a finite cover admitting an infinite trivalent tree of geometric ideal triangulations. Furthermore, every sufficiently long Dehn filling of this cover also admits infinitely many geometric ideal triangulations.

The proof involves a mixture of geometric constructions and subgroup separability tools. One of the separability tools is a new theorem about separating a peripheral subgroup from every conjugate of a coset. I will try to give a glimpse into both the geometry and the subgroup separability. This is joint work with Emily Hamilton and Neil Hoffman.