On Isolated Geometric Triangulations

Work of Kalelkar, Schleimer, and Segerman shows that, with some exceptions, the set of essential ideal triangulations of an orientable cusped hyperbolic 3-manifold is connected via 2-3 and 3-2 moves. It is natural to ask if the subgraph consisting of only those triangulations that are geometric is connected. Hoffman gives the first two examples of geometric triangulations with the property that no 2-3 or 3-2 move results in a geometric triangulation. In this paper, we introduce these as isolated geometric triangulations and show that this is not a property of small manifolds by exhibiting an infinite family of once-punctured torus bundles whose monodromy ideal triangulation is isolated.