Infinitely many virtual geometric triangulations

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, long Dehn filling of one cusp in this admits This is constructed several stages, using results about separability peripheral subgroups and their double cosets, addition to new conjugacy theorem may be independent interest. The infinite sequence triangulations supported submanifold associated cusp, can organized into an trivalent tree Pachner moves.