Differential Geometry
Growth tightness of negatively curved manifolds
[Croissance forte des variétés à courbure négative]
Comptes Rendus. Mathématique, Tome 336 (2003) no. 6, pp. 487-491.

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.

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.

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Accepté le :
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DOI : 10.1016/S1631-073X(03)00086-4
Sambusetti, Andrea 1

1 Instituto “G. Castelnuovo”, Università “La Sapienza”, P.le A. Moro 5, 00185 Roma, Italy
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Sambusetti, Andrea. Growth tightness of negatively curved manifolds. Comptes Rendus. Mathématique, Tome 336 (2003) no. 6, pp. 487-491. doi : 10.1016/S1631-073X(03)00086-4. http://archive.numdam.org/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] Grigorchuk, R.; De La Harpe, P. 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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