Length functions on mapping class groups and simplicial volumes of mapping tori
Annales de l'Institut Fourier, Online first, 24 p.

Let M be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group MCG(M) of M, which are defined in terms of filling norms on the space of singular boundaries on M, both with real and with integral coefficients. We show that filling volumes are length functions on MCG(M), we prove that the real filling volume of a mapping class f is equal to the simplicial volume of the corresponding mapping torus E f , while the integral filling volume of f is not smaller than the stable integral simplicial volume of E f .

We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of 3-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.

Soit M une variété fermée et orientable. Nous introduisons deux invariants numériques, dénotés volumes de remplissage, sur le groupe modulaire MCG(M) de M. Les invariants sont définis en fonction de normes de remplissage sur l’espace des bords singuliers de M, 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 f est égal au volume simplicial du tore d’application correspondant E f , et que le volume de remplissage entier de f n’est pas plus petit que le volume simplicial entier stable de E f .

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.

Online First:
DOI: 10.5802/aif.3610
Classification: 55N10, 57S05, 53C23, 57M07
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

1 Scuola Normale Superiore, Pisa (Italy)
2 Dipartimento di Matematica, Università di Pisa, (Italy)
     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},
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Bertolotti, Federica; Frigerio, Roberto. Length functions on mapping class groups and simplicial volumes of mapping tori. Annales de l'Institut Fourier, Online first, 24 p.

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