Generalized Mom-structures and ideal triangulations of 3-manifolds with non-spherical boundary

TL;DR

This paper extends Mom-structures to 3-manifolds with non-spherical boundary, relating them to ideal triangulations and providing moves to connect different structures, enhancing understanding of 3-manifold decompositions.

Contribution

It introduces generalized protoMom-structures for manifolds with boundary and establishes their relation to ideal triangulations and geometric decompositions.

Findings

Relation between protoMom-structures and ideal triangulations

Construction of geometric protoMom-structures from decompositions

Set of moves connecting any two protoMom-structures

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 non-empty 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 non-closed hyperbolic manifolds without annular cusps, we describe how an internal geometric protoMom-structure can be constructed starting from 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.