We show that any two geometric triangulations of a hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes and upper and lower bounds on the lengths of edges of the triangulation. This gives an algorithm to check if two geometrically triangulated compact hyperbolic or low dimensional spherical manifolds are isometric.

We show that geometric triangulations of a compact hyperbolic, spherical or Euclidean n-manifold are related by geometric Pachner moves up to barycentric subdivisions. For n ≤ 3, we show that geometric triangulations of a closed hyperbolic, spherical or Euclidean n-manifold are related by geometric Pachner moves.

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.

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