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The triangulation complexity of fibred $3$–manifolds

Marc Lackenby and Jessica S Purcell

Geometry & Topology 28 (2024) 1727–1828
Abstract

The triangulation complexity of a closed orientable 3–manifold M is the minimal number of tetrahedra in any triangulation of M. Our main theorem gives upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3–manifold that fibres over the circle. We show that the triangulation complexity of the manifold is equal to the translation length of the monodromy action on the mapping class group of the fibre S, up to a bounded factor, where the bound depends only on the genus of S.

Keywords
triangulation complexity, $3$–manifold, fibred, pseudo-Anosov
Mathematical Subject Classification
Primary: 57K20, 57K30, 57Q15
Secondary: 57K32
References
Publication
Received: 6 April 2021
Revised: 21 July 2022
Accepted: 30 September 2022
Published: 18 July 2024
Proposed: Ian Agol
Seconded: Mladen Bestvina, David Fisher
Authors
Marc Lackenby
Mathematical Institute
University of Oxford
Oxford
United Kingdom
Jessica S Purcell
School of Mathematics
Monash University
Melbourne VIC
Australia

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