Comptes Rendus
Differential Geometry
Growth tightness of negatively curved manifolds
Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 487-491.

We show that any closed negatively curved manifold X is growth tight: this means that its universal covering X ˜ has an exponential growth rate ω(X ˜) which is strictly greater than the exponential growth rate ω(X ¯) of any other normal covering X ¯. Moreover, we give an explicit formula which estimates the difference between ω(X ˜) and ω(X ¯) in terms of the systole of X ¯ 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 X ˜ a un taux de croissance exponentielle ω(X ˜) strictement supérieur à celui de n'importe quel autre revêtement normal X ¯ de X. Plus précisément, on donne une formule estimant explicitement la différence entre ces taux de croissance, ω(X ˜) et ω(X ¯), en termes de la systole de X ¯ 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.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00086-4

Andrea Sambusetti 1

1 Instituto “G. Castelnuovo”, Università “La Sapienza”, P.le A. Moro 5, 00185 Roma, Italy
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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/

[1] G.N. Arzhantseva, I.G. Lysenok, Growth tightness for word hyperbolic groups, Prépublication de l'Université de Genève, 2001

[2] R. Grigorchuk; P. De La Harpe On problems related to growth, entropy and spectrum in group theory, J. Dynam. Control Systems, Volume 3 (1997) no. 1, pp. 51-89

[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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