Volume-convergent sequences of Haken 3-manifolds

P. Derbez

doi:10.1016/S1631-073X(03)00187-0

Topology

Volume-convergent sequences of Haken 3-manifolds
[Suites de variétés Haken dont les volumes convergent]

Présenté par : Étienne Ghys

P. Derbez ¹

¹ Université de Bourgogne, U.F.R. sciences et techniques, 9, avenue Alain Savary, BP 47870, 21078 Dijon cedex, France

Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 833-838.

Résumé
Abstract

Soit M une 3-variété close orientable et désignons par Vol(M) le volume simplicial de Gromov de M. Cette Note est consacrée à l'étude des applications de degré non-nul $f_{i} : M \to N_{i}$ où chaque N_i est une variété Haken. Le résultat principal affirme que toute suite (N_i,f_i) de variétés Haken satisfaisant lim_i→∞deg(f_i)×Vol(N_i)=Vol(M) est finie, à homéomorphisme près. Ce résultat implique en particulier que toute 3-variété close orientable dont le volume simplicial de Gromov est nul (en particulier toute variété graphée) domine au plus un nombre fini de variétés Haken.

Let M be a closed orientable 3-manifold and let Vol(M) denote its Gromov simplicial volume. This paper is devoted to the study of sequences of non-zero degree maps $f_{i} : M \to N_{i}$ to Haken manifolds. We prove that any sequence of Haken manifolds (N_i,f_i), satisfying lim_i→∞deg(f_i)×Vol(N_i)=Vol(M) is finite up to homeomorphism. As an application, we deduce from this fact that any closed orientable 3-manifold with zero Gromov simplicial volume and in particular any graph manifold dominates at most finitely many Haken 3-manifolds.

Reçu le : 2003-01-14
Accepté le : 2003-03-30
Publié le : 2003-05-08

DOI : 10.1016/S1631-073X(03)00187-0

Affiliations des auteurs :

P. Derbez ¹

¹ Université de Bourgogne, U.F.R. sciences et techniques, 9, avenue Alain Savary, BP 47870, 21078 Dijon cedex, France

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     title = {Volume-convergent sequences of {Haken} 3-manifolds},
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P. Derbez. Volume-convergent sequences of Haken 3-manifolds. Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 833-838. doi : 10.1016/S1631-073X(03)00187-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00187-0/

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