An Upper Bound on Pachner Moves Relating Geometric Triangulations

Abstract

Suppose that M is a compact, connected three-manifold with boundary. We show that if the universal cover has infinitely many boundary components then M has an ideal triangulation which is essential: no edge can be homotoped into the boundary. Under the same hypotheses, we show that the set of essential triangulations of M is connected via 2-3, 3-2, 0-2, and 2-0 moves.The above results are special cases of our general theory. We introduce L-essential triangulations: boundary components of the universal cover receive labels and no edge has the same label at both ends. As an application, under mild conditions on a representation, we construct an ideal triangulation for which a solution to Thurston's gluing equations recovers the given representation.Our results also imply that such triangulations are connected via 2-3, 3-2, 0-2, and 2-0 moves. Together with results of Pandey and Wong, this proves that Dimofte and Garoufalidis' 1-loop invariant is independent of the choice of essential triangulation.