We show that any closed negatively curved manifold X is growth tight: this means that its universal covering has an exponential growth rate which is strictly greater than the exponential growth rate of any other normal covering . Moreover, we give an explicit formula which estimates the difference between and in terms of the systole of and of some geometric parameters of the base manifold X. Then, we describe some applications to systoles and periodic geodesics.
On montre que toute variété fermée X de courbure négative est à croissance forte : cela signifie que le revêtement universel a un taux de croissance exponentielle strictement supérieur à celui de n'importe quel autre revêtement normal de X. Plus précisément, on donne une formule estimant explicitement la différence entre ces taux de croissance, et , en termes de la systole de et d'autres simples paramètres géométriques de la variété de base X. On en déduit ensuite une inégalité systolique et une application aux géodésiques périodiques.
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Andrea Sambusetti 1
@article{CRMATH_2003__336_6_487_0, author = {Andrea Sambusetti}, title = {Growth tightness of negatively curved manifolds}, journal = {Comptes Rendus. Math\'ematique}, pages = {487--491}, publisher = {Elsevier}, volume = {336}, number = {6}, year = {2003}, doi = {10.1016/S1631-073X(03)00086-4}, language = {en}, }
Andrea Sambusetti. Growth tightness of negatively curved manifolds. Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 487-491. doi : 10.1016/S1631-073X(03)00086-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00086-4/
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[3] A. Sambusetti, Growth tightness of free and amalgamated products, Ann. Sci. École Norm. Sup., to appear
[4] A. Sambusetti, Growth tightness of surface groups, Exposition. Math., to appear
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