Volume-convergent sequences of Haken 3-manifolds

Derbez, P.

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

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 $f_i:\mathrm{M}\to {\mathrm{N}}_i$ 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 $f_i:\mathrm{M}\to {\mathrm{N}}_i$ 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.

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