Connecting essential triangulations

Let M be a 3-dimensional compact manifold with non-empty boundary. There exists a collection of tetrahedra stuck in pairs along their faces such that deleting all their vertices gives a space that is homeomorphic to the interior of M. This is called a (ideal) triangulation of M. Amendola, Matveev and Piergallini have shown that any two such triangulations of M are connected by a sequence of triangulations, where each triangulation in the sequence is obtained from the previous one by a local combinatorial change, namely a 2-3 or 3-2 move. For some applications however, we need our triangulations to have special properties. We call a triangulation essential if none of its edge is a homotopically trivial loop. Certain essential triangulations are isolated, in the sense that any 2-3 or 3-2 move introduces an inessential edge. We have shown that when the universal cover of M has infinitely many boundary components and the two given essential triangulations of M are not isolated, then they are connected by a sequence of 2-3 and 3-2 moves through essential triangulations. If in addition, we allow V-moves and their inverses, then any two essential triangulations (including the isolated ones) are connected via essential triangulations. Our results have applications such as proving that the 1-loop invariant is independent of the choice of triangulation. This is joint work with Saul Schleimer and Henry Segerman.