[Fonctions de longueur sur les groupes modulaires et volume simplicial de tores d’application]
Soit une variété fermée et orientable. Nous introduisons deux invariants numériques, dénotés volumes de remplissage, sur le groupe modulaire de . Les invariants sont définis en fonction de normes de remplissage sur l’espace des bords singuliers de , avec coefficients réels ou entiers. Nous montrons que les volumes de remplissage sont des fonctions de longueur, nous prouvons que le volume de remplissage réel d’une classe est égal au volume simplicial du tore d’application correspondant , et que le volume de remplissage entier de n’est pas plus petit que le volume simplicial entier stable de .
Nous exposons plusieurs résultats d’annulation et de positivité des volumes de remplissage. Comme conséquence, nous montrons que le volume hyperbolique des tores d’application tridimensionnels n’est pas sous-additif par rapport à la monodromie, et que les normes de remplissage réelle et entiére sur les bords entiers ne sont souvent pas biLipschitz-équivalents.
Let be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group of , which are defined in terms of filling norms on the space of singular boundaries on , both with real and with integral coefficients. We show that filling volumes are length functions on , we prove that the real filling volume of a mapping class is equal to the simplicial volume of the corresponding mapping torus , while the integral filling volume of is not smaller than the stable integral simplicial volume of .
We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of -dimensional mapping tori is not subadditive with respect to their monodromy, and that the real and the integral filling norms on integral boundaries are often non-biLipschitz equivalent.
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Keywords: Simplicial volume, Stable integral simplicial volume, Filling volume, Length functions, Mapping torus, fibration over the circle, Mapping class group.
Mot clés : Volume simplicial, volume simplicial entier stable, volume de remplissage, fonctions de longueur, tore d’application, groupe modulaire.
Bertolotti, Federica 1 ; Frigerio, Roberto 2
CC-BY-ND 4.0
@article{AIF_2024__74_4_1383_0,
author = {Bertolotti, Federica and Frigerio, Roberto},
title = {Length functions on mapping class groups and simplicial volumes of mapping tori},
journal = {Annales de l'Institut Fourier},
pages = {1383--1406},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {74},
number = {4},
year = {2024},
doi = {10.5802/aif.3610},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3610/}
}
TY - JOUR AU - Bertolotti, Federica AU - Frigerio, Roberto TI - Length functions on mapping class groups and simplicial volumes of mapping tori JO - Annales de l'Institut Fourier PY - 2024 SP - 1383 EP - 1406 VL - 74 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3610/ DO - 10.5802/aif.3610 LA - en ID - AIF_2024__74_4_1383_0 ER -
%0 Journal Article %A Bertolotti, Federica %A Frigerio, Roberto %T Length functions on mapping class groups and simplicial volumes of mapping tori %J Annales de l'Institut Fourier %D 2024 %P 1383-1406 %V 74 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3610/ %R 10.5802/aif.3610 %G en %F AIF_2024__74_4_1383_0
Bertolotti, Federica; Frigerio, Roberto. Length functions on mapping class groups and simplicial volumes of mapping tori. Annales de l'Institut Fourier, Tome 74 (2024) no. 4, pp. 1383-1406. doi : 10.5802/aif.3610. https://aif.centre-mersenne.org/articles/10.5802/aif.3610/
[1] Hyperplane arrangements in negatively curved manifolds and relative hyperbolicity, Groups Geom. Dyn., Volume 7 (2013) no. 1, pp. 13-38 | DOI | MR | Zbl
[2] Self-homotopy equivalences of 3-Manifolds (2022) (https://arxiv.org/abs/2207.05717)
[3] The simplicial volume of mapping tori of 3-manifolds, Math. Ann., Volume 376 (2020) no. 3-4, pp. 1429-1447 | DOI | MR | Zbl
[4] Some remarks on the simplicial volume of nonpositively curved manifolds, Math. Ann., Volume 377 (2020) no. 3-4, pp. 969-987 | DOI | MR | Zbl
[5] Variations on the theme of the uniform boundary condition, J. Topol. Anal., Volume 13 (2021) no. 1, pp. 147-174 | DOI | MR | Zbl
[6] Stable integral simplicial volume of 3-manifolds, J. Topol., Volume 14 (2021) no. 2, pp. 608-640 | DOI | MR | Zbl
[7] Bounded cohomology of discrete groups, Mathematical Surveys and Monographs, 227, American Mathematical Society, 2017, xvi+193 pages | DOI | MR | Zbl
[8] Gromov’s theory of multicomplexes with applications to bounded cohomology and simplicial volume, Memoirs of the American Mathematical Society, 283, American Mathematical Society, 2023 no. 1402, vi+153 pages | DOI | MR | Zbl
[9] An example of a closed 5-manifold of nonpositive curvature that fibers over a circle (2021) (https://arxiv.org/abs/2106.08549, to appear in Groups Geom. Dyn.)
[10] Volume and bounded cohomology, Publ. Math., Inst. Hautes Étud. Sci. (1982) no. 56, pp. 5-99 | Numdam | MR | Zbl
[11] Hyperbolic groups, Essays in group theory (Mathematical Sciences Research Institute Publications), Volume 8, Springer, 1987, pp. 75-263 | DOI | MR | Zbl
[12] Simplices of maximal volume in hyperbolic -space, Acta Math., Volume 147 (1981) no. 1-2, pp. 1-11 | DOI | MR | Zbl
[13] Hyperbolic 5-manifolds that fiber over , Invent. Math., Volume 231 (2023) no. 1, pp. 1-38 | DOI | MR | Zbl
[14] Simplicial volume and essentiality of manifolds fibered over spheres, J. Topol., Volume 16 (2023) no. 1, pp. 192-206 | DOI | MR | Zbl
[15] Entropy versus volume for pseudo-Anosovs, Exp. Math., Volume 18 (2009) no. 4, pp. 397-407 | DOI | MR | Zbl
[16] Entropy, Weil–Petersson translation distance and Gromov norm for surface automorphisms, Proc. Am. Math. Soc., Volume 140 (2012) no. 11, pp. 3993-4002 | DOI | MR | Zbl
[17] Normalized entropy versus volume for pseudo-Anosovs, Geom. Topol., Volume 22 (2018) no. 4, pp. 2403-2426 | DOI | MR | Zbl
[18] The triangulation complexity of fibred 3-manifolds (2019) (https://arxiv.org/abs/1910.10914, to appear in Geom. Topol.)
[19] Homological filling functions with coefficients, Groups Geom. Dyn., Volume 16 (2022) no. 3, pp. 889-907 | DOI | MR | Zbl
[20] Homology and simplicial volume, Ph. D. Thesis, WWU Münster (2007) (available online at http://nbn-resolving.de/urn:nbn:de:hbz:6-37549578216)
[21] Simplicial volume with -coefficients, Period. Math. Hung., Volume 80 (2020) no. 1, pp. 38-58 | DOI | MR | Zbl
[22] Bounded cohomology of amenable covers via classifying spaces, Enseign. Math., Volume 66 (2020) no. 1-2, pp. 151-172 | DOI | MR | Zbl
[23] Bounded cohomology of certain groups of homeomorphisms, Proc. Am. Math. Soc., Volume 94 (1985) no. 3, pp. 539-544 | DOI | MR | Zbl
[24] Simplices of maximal volume or minimal total edge length in hyperbolic space, J. Lond. Math. Soc., Volume 66 (2002) no. 3, pp. 753-768 | DOI | MR | Zbl
[25] Homogeneous length functions on groups, Algebra Number Theory, Volume 12 (2018) no. 7, pp. 1773-1786 | DOI | MR | Zbl
[26] Surface bundles over which are -fold branched cyclic coverings of , Math. Sem. Notes Kobe Univ., Volume 9 (1981) no. 1, pp. 159-180 | MR | Zbl
[27] Volume and homology growth of aspherical manifolds, Geom. Topol., Volume 20 (2016) no. 2, pp. 1035-1059 | DOI | MR | Zbl
[28] Length functions on groups and rigidity (2021) (https://arxiv.org/abs/2101.08902)
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