Generalized Mom-structures and ideal triangulations of $3$–manifolds with nonspherical boundary

Abstract

The so-called Mom-structures on hyperbolic cusped $3$–manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of $3$–manifolds with nonempty boundary that does not have spherical components. We then describe a certain relation between such generalized Mom-structures, called protoMom-structures, internal on a fixed $3$–manifold $N$, and ideal triangulations of $N$; in addition, in the case of nonclosed hyperbolic manifolds without annular cusps, we describe how an internal geometric protoMom-structure can be constructed starting from the Epstein–Penner or Kojima decomposition. Finally, we exhibit a set of combinatorial moves that relate any two internal protoMom-structures on a fixed $N$ to each other.