The triangulation complexity of fibred 3-manifolds (Q6595800)

For a closed orientable 3-manifold \(M\) the triangulation complexity \(\Delta (M)\) is the minimal number of tetrahedra in any triangulation of \(M\). See also the complexity theory of compact 3-manifolds developed by \textit{S. Matveev} [Algorithmic topology and classification of 3-manifolds. Berlin: Springer (2003; Zbl 1048.57001)], where the complexity is the minimal number of vertices of a simple spine of \(M\). An important property is that only finitely many 3-manifolds have triangulation complexity less than a given number. Up to now, precise values for \(\Delta(M)\) are known for a few relatively small examples and for a few infinite families of 3-manifolds. There is a natural relation between the triangulation complexity of a hyperbolic 3-manifold \(M\) and its volume \(\operatorname{vol} (M)\). Namely, according to Gromov and Thurston \(\Delta(M)\) is at least \(\operatorname{vol} (M) / v_3\), where \(v_3\approx1.01494\) is the volume of a regular hyperbolic ideal 3-simplex.\N\NIn this paper the authors consider 3-manifolds that fiber over the circle and show that triangulation complexity is nicely related to many other important topological and geometric quantities. The main theorem (Theorem~1.3) relates the triangulation complexity \(\Delta(M)\) with the discrete geometry of the mapping class group of \(S\) and with the Teichmüller space of \(S\) as following. Let \(S\) be a closed orientable surface of genus at least two, and let \(\phi: S \to S\) be a pseudo-Anosov homeomorphism. Then the following quantities are within bounded ratios of each other, where the bounds only depend on the genus of \(S\) and a choice of finite generating set for the mapping class group of \(S\): (1) the triangulation complexity of \((S \times I) / \phi\); (2) the translation length of \(\phi\) in the thick part of the Teichmüller space of \(S\); (3) the translation length of \(\phi\) in the mapping class group of \(S\); (4) the stable translation length of \(\phi\) in the mapping class groups of \(S\).\N\NTheorem 1.4 describes the product situation as following. Let \(S\) be a closed orientable surface with genus at least two and let \(T_0\) and \(T_1\) be non-isotopic 1-vertex triangulations of \(S\). Then the following are within a bounded ration of each other, the bound only depending on the genus of \(S\): (1) the minimal number of tetrahedra in any triangulation \(T\) of \(S\times [0,1]\) such that the restriction of \(T\) to \(S\times\{0\}\) equals \(T_0\) and the restriction of \(T\) to \(S \times \{1\}\) equals \(T_1\); (2) the minimal number of 2-2 Pachner moves relating \(T_0\) and \(T_1\).\N\NThe method introduced to prove the main result is involving various ideas and is interesting and useful itself. That admitted the authors to obtain the analogous results for elliptic and Sol 3-manifolds in [\textit{M. Lackenby} and \textit{J. S. Purcell}, Math. Ann. 390, No. 2, 1623--1667 (2024; Zbl 07932344)].\N\NFinally, the authors formulate the following conjecture (Conjecture~1.7). Let \(M\) be a closed orientable manifold with a Heegaard surface \(S\) that divides \(M\) into handlebodies \(V\) and \(W\). Then the complexity \(\Delta(M)\) is bounded above and below by linear functions of the distance in the spine graph \(Sp(S)\) between the disc subsets \(D_V\) and \(D_W\).