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