The triangulation complexity of fibred 3–manifolds

Lackenby, Marc¹ ; Purcell, Jessica S²

¹ Mathematical Institute, University of Oxford, Oxford, United Kingdom
² School of Mathematics, Monash University, Melbourne VIC, Australia

Geometry & topology, Tome 28 (2024) no. 4, pp. 1727-1828.

Résumé

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 $S$.

Détail

DOI : 10.2140/gt.2024.28.1727

Keywords: triangulation complexity, $3$–manifold, fibred, pseudo-Anosov

Bibliographie

[1] I Agol, Ideal triangulations of pseudo-Anosov mapping tori, from: "Topology and geometry in dimension three" (editors W Li, L Bartolini, J Johnson, F Luo, R Myers, J H Rubinstein), Contemp. Math. 560, Amer. Math. Soc. (2011) 1 | DOI

[2] M Bell, flipper, software (2019)

[3] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[4] J F Brock, The Weil–Petersson metric and volumes of 3–dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003) 495 | DOI

[5] J F Brock, Weil–Petersson translation distance and volumes of mapping tori, Comm. Anal. Geom. 11 (2003) 987 | DOI

[6] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, 9, Cambridge Univ. Press (1988) | DOI

[7] K Culik Ii, D Wood, A note on some tree similarity measures, Inform. Process. Lett. 15 (1982) 39 | DOI

[8] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[9] A Fathi, F Laudenbach, V Poénaru, Thurston’s work on surfaces, 48, Princeton Univ. Press (2012)

[10] D Futer, S Schleimer, Cusp geometry of fibered 3–manifolds, Amer. J. Math. 136 (2014) 309 | DOI

[11] U Hamenstädt, Geometry of the mapping class groups, I : Boundary amenability, Invent. Math. 175 (2009) 545 | DOI

[12] W Jaco, U Oertel, An algorithm to decide if a 3–manifold is a Haken manifold, Topology 23 (1984) 195 | DOI

[13] W Jaco, H Rubinstein, S Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009) 157 | DOI

[14] W Jaco, J H Rubinstein, S Tillmann, Coverings and minimal triangulations of 3–manifolds, Algebr. Geom. Topol. 11 (2011) 1257 | DOI

[15] W Jaco, J H Rubinstein, S Tillmann, Z2–Thurston norm and complexity of 3–manifolds, Math. Ann. 356 (2013) 1 | DOI

[16] W Jaco, J H Rubinstein, J Spreer, S Tillmann, Z2–Thurston norm and complexity of 3–manifolds, II, Algebr. Geom. Topol. 20 (2020) 503 | DOI

[17] M Lackenby, Taut ideal triangulations of 3–manifolds, Geom. Topol. 4 (2000) 369 | DOI

[18] M Lackenby, The crossing number of composite knots, J. Topol. 2 (2009) 747 | DOI

[19] M Lackenby, J S Purcell, The triangulation complexity of lens spaces and sol manifolds, Math. Ann. (2024) | DOI

[20] W B R Lickorish, A finite set of generators for the homeotopy group of a 2–manifold, Proc. Cambridge Philos. Soc. 60 (1964) 769 | DOI

[21] B Martelli, C Petronio, Three-manifolds having complexity at most 9, Exp. Math. 10 (2001) 207 | DOI

[22] H Masur, L Mosher, S Schleimer, On train-track splitting sequences, Duke Math. J. 161 (2012) 1613 | DOI

[23] S Matveev, Algorithmic topology and classification of 3–manifolds, 9, Springer (2003) | DOI

[24] S V Matveev, Tabulation of three-dimensional manifolds, Uspekhi Mat. Nauk 60 (2005) 97

[25] S Matveev, C Petronio, A Vesnin, Two-sided asymptotic bounds for the complexity of some closed hyperbolic three-manifolds, J. Aust. Math. Soc. 86 (2009) 205 | DOI

[26] R C Penner, J L Harer, Combinatorics of train tracks, 125, Princeton Univ. Press (1992) | DOI

[27] C Petronio, A Vesnin, Two-sided bounds for the complexity of cyclic branched coverings of two-bridge links, Osaka J. Math. 46 (2009) 1077

[28] M Stocking, Almost normal surfaces in 3–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171 | DOI

[29] A Thompson, Thin position and the recognition problem for S3, Math. Res. Lett. 1 (1994) 613 | DOI

[30] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417 | DOI

[31] W P Thurston, Collected works of William P Thurston with commentary, IV: The geometry and topology of three-manifolds, Amer. Math. Soc. (2022)