Length functions on mapping class groups and simplicial volumes of mapping tori

Bertolotti, Federica; Frigerio, Roberto

doi:10.5802/aif.3610

Length functions on mapping class groups and simplicial volumes of mapping tori
[Fonctions de longueur sur les groupes modulaires et volume simplicial de tores d’application]

Bertolotti, Federica ¹ ; Frigerio, Roberto ²

¹ Scuola Normale Superiore, Pisa (Italy)
² Dipartimento di Matematica, Università di Pisa, (Italy)

Annales de l'Institut Fourier, Online first, 24 p.

Résumé
Abstract

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.

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.

Reçu le : 2022-06-14
Révisé le : 2022-09-18
Accepté le : 2022-11-10
Première publication : 2024-05-17

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.

Affiliations des auteurs :

Bertolotti, Federica ¹ ; Frigerio, Roberto ²

¹ Scuola Normale Superiore, Pisa (Italy)
² Dipartimento di Matematica, Università di Pisa, (Italy)

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     title = {Length functions on mapping class groups and simplicial volumes of mapping tori},
     journal = {Annales de l'Institut Fourier},
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     doi = {10.5802/aif.3610},
     language = {en},
     note = {Online first},
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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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