An 1 -norm-mass inequality for complete manifolds
Annales de l'Institut Fourier, Online first, 22 p.

We generalize an inequality of Besson–Courtois–Gallot about volume and simplicial volume of closed manifolds to the 1 -norm of all the homology classes of complete manifolds. The inequality involves the critical exponent of the fundamental group of the manifold and the mass of the homology classes.

Nous généralisons une inégalité de Besson–Courtois–Gallot entre le volume et le volume simplicial des variétés fermées à la norme 1 de toutes les classes d’homologie des variétés complètes. L’inégalité s’exprime en termes de l’exposant critique du groupe fondamental de la variété et de la masse des classes d’homologie.

Received:
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Accepted:
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DOI: 10.5802/aif.3643
Classification: 53C23, 57R57
Keywords: Gromov norm, critical exponent, mass, comass, complete manifold.
Mot clés : norme de Gromov, exposant critique, masse, comasse, variété complète.
Campagnolo, Caterina 1; Wang, Shi 2

1 Universidad Autónoma de Madrid Departamento de Matemáticas C/ Nicolás Cabrera, 13-15 28049 Madrid (Spain)
2 ShanghaiTech University Institute of Mathematical Sciences Pudong, Shanghai (China)
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Campagnolo, Caterina; Wang, Shi. An $\ell _1$-norm-mass inequality for complete manifolds. Annales de l'Institut Fourier, Online first, 22 p.

[1] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Springer, 2012 | Zbl

[2] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Volume et entropie minimale des espaces localement symétriques, Invent. Math., Volume 103 (1991) no. 2, pp. 417-445 | DOI | MR | Zbl

[3] Bucher, Michelle Simplicial volume of products and fiber bundles, Discrete groups and geometric structures (Contemporary Mathematics), Volume 501, American Mathematical Society, 2009, pp. 79-86 | DOI | MR | Zbl

[4] Bucher, Michelle; Monod, Nicolas The norm of the Euler class, Math. Ann., Volume 353 (2012) no. 2, pp. 523-544 | DOI | MR | Zbl

[5] Bucher-Karlsson, Michelle The simplicial volume of closed manifolds covered by 2 × 2 , J. Topol., Volume 1 (2008) no. 3, pp. 584-602 | DOI | MR | Zbl

[6] Burger, Marc; Iozzi, Alessandra; Wienhard, Anna Surface group representations with maximal Toledo invariant, Ann. Math., Volume 172 (2010) no. 1, pp. 517-566 | DOI | MR | Zbl

[7] Connell, Chris; Wang, Shi Positivity of simplicial volume for nonpositively curved manifolds with a Ricci-type curvature condition, Groups Geom. Dyn., Volume 13 (2019) no. 3, pp. 1007-1034 | DOI | MR | Zbl

[8] Crowley, Diarmuid; Löh, Clara Functorial seminorms on singular homology and (in)flexible manifolds, Algebr. Geom. Topol., Volume 15 (2015) no. 3, pp. 1453-1499 | DOI | MR | Zbl

[9] Frigerio, Roberto Bounded cohomology of discrete groups, Mathematical Surveys and Monographs, 227, American Mathematical Society, 2017, xvi+193 pages | DOI | MR | Zbl

[10] Frigerio, Roberto; Moraschini, Marco Gromov’s theory of multicomplexes with applications to bounded cohomology and simplicial volume, Mem. Am. Math. Soc., Volume 283 (2023) no. 1402, p. vi+153 | DOI | MR | Zbl

[11] Gromov, Michael Volume and bounded cohomology, Publ. Math., Inst. Hautes Étud. Sci. (1982) no. 56, p. 5-99 (1983) | Numdam | MR | Zbl

[12] Haagerup, Uffe; Munkholm, Hans J. Simplices of maximal volume in hyperbolic n-space, Acta Math., Volume 147 (1981) no. 1-2, pp. 1-11 | DOI | MR | Zbl

[13] Inoue, Hisao; Yano, Koichi The Gromov invariant of negatively curved manifolds, Topology, Volume 21 (1982) no. 1, pp. 83-89 | DOI | MR | Zbl

[14] Johnson, Barry E. Cohomology in Banach algebras, Memoirs of the American Mathematical Society, 127, American Mathematical Society, 1972, iii+96 pages | DOI | MR | Zbl

[15] Kapovich, Michael Homological dimension and critical exponent of Kleinian groups, Geom. Funct. Anal., Volume 18 (2009) no. 6, pp. 2017-2054 | DOI | MR | Zbl

[16] Lafont, Jean-François; Schmidt, Benjamin Simplicial volume of closed locally symmetric spaces of non-compact type, Acta Math., Volume 197 (2006) no. 1, pp. 129-143 | DOI | MR | Zbl

[17] Lafont, Jean-François; Wang, Shi Barycentric straightening and bounded cohomology, J. Eur. Math. Soc., Volume 21 (2019) no. 2, pp. 381-403 | DOI | MR | Zbl

[18] Löh, Clara; Sauer, Roman Simplicial volume of Hilbert modular varieties, Comment. Math. Helv., Volume 84 (2009) no. 3, pp. 457-470 | DOI | MR | Zbl

[19] Mineyev, Igor Bounded cohomology characterizes hyperbolic groups, Q. J. Math., Volume 53 (2002) no. 1, pp. 59-73 | DOI | MR | Zbl

[20] Morita, Shigeyuki Characteristic classes of surface bundles and bounded cohomology, A fête of topology, Academic Press Inc., 1988, pp. 233-257 | DOI | MR | Zbl

[21] Munkholm, Hans J. Simplices of maximal volume in hyperbolic space, Gromov’s norm, and Gromov’s proof of Mostow’s rigidity theorem (following Thurston), Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979) (Lecture Notes in Mathematics), Volume 788, Springer (1980), pp. 109-124 | DOI | MR | Zbl

[22] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, 2016, xviii+499 pages | DOI | MR | Zbl

[23] Pieters, Hester New bounds for the simplicial volume of complex hyperbolic surfaces (2018) (https://arxiv.org/abs/1812.11541)

[24] Thurston, William P. The geometry and topology of three-manifolds, Princeton University, 1979

[25] Wang, Shi On splitting rank of non-compact-type symmetric spaces and bounded cohomology, J. Topol. Anal., Volume 12 (2020) no. 2, pp. 465-489 | DOI | MR | Zbl

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