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

Ekaterina Pervova

Algebraic & Geometric Topology 12 (2012) 235–265

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.

$3$–manifold, triangulation, Mom-structure
Mathematical Subject Classification 2010
Primary: 57M20, 57N10
Secondary: 57M15, 57M50
Received: 17 March 2011
Revised: 24 August 2011
Accepted: 7 September 2011
Published: 25 February 2012
Ekaterina Pervova
Dipartimento di Matematica Applicata
University of Pisa
Via F Buonarroti 1C
I-56127 Pisa