Uniform growth in hyperbolic groups. (Croissance uniforme dans les groupes hyperboliques.) (French) Zbl 0914.20033

Let \(G\) be a group of finite type and let \(S\) be a symmetric set of generators of \(G\). The exponential growth rate \(\tau(G,S)\) is defined as \[ \tau(G,S)=\limsup_{n\to\infty}(\text{Card}(B(n)))^{1/2}. \] The author proves the following Theorem: Let \(G\) be a word hyperbolic group which is non-elementary. Then, there exists a constant \(c_G>1\) such that for any symmetric generating system \(S\), we have \(\tau(G,S)\geq c_G\). As the author notes, in the case where \(G\) is torsion free, the result had already been proved by Grigorchuk and de la Harpe. The author uses for the proof of his theorem the same outline as that of the result of Grigorchuk and de la Harpe.
The author proves also the following Proposition: Let \(G\) be an infinite word hyperbolic group. Then there exists an integer \(M_G\) such that for any generating system \(S\) of \(G\), the group \(G\) equipped with the word metric associated to \(S\) contains a hyperbolic element of length \(\leq M_G\).


20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
57M07 Topological methods in group theory
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[1] [1] , , , Géométrie et théorie des groupes, Lecture Notes in Mathematics, 1441 (1990). · Zbl 0727.20018
[2] [2] , Sous-groupes à deux générateurs des groupes hyperboliques, Group Theory from a Geometrical Viewpoint, World Scientific, 1990, 177-189. · Zbl 0845.20027
[3] [3] , Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J., 83, Vol. 3 (Juin 1996), 661-682. · Zbl 0852.20032
[4] [4] , , Sur les groupes hyperboliques d’après M. Gromov, Birkhäuser, Boston, 1990. · Zbl 0731.20025
[5] [5] , On growth in group theory, Proceedings of the International Congress of Mathematicians, vol. I, II (Kyoto, 1990), 325-338. · Zbl 0749.20016
[6] [6] , , On finitely generated groups and problems related to growth, Preprint, Genève, Novembre 1996. · Zbl 1004.20018
[7] [7] , Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer-Verlag, New-York, 1987, 75-263. · Zbl 0634.20015
[8] [8] , , Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc., 348, vol 6 (1996), 2091-2138. · Zbl 0876.20023
[9] [9] , , Combinatorial group theory, Springer, 1977. · Zbl 0368.20023
[10] [10] , Points fixes des automorphismes de groupe hyperbolique, Annales de l’Institut Fourier, 39-3 (1989), 651-662. · Zbl 0674.20022
[11] [11] , Invariants topologiques et géométriques reliés aux longueurs des géodésiques et aux sections harmoniques de fibrés, Thèse Institut Fourier (Grenoble), Octobre 1994.
[12] [3] , Sur la méthode des orbites. Proceedings de la conférence : «Non commutative Harmonic Analysis», Marseille-Luminy, Lecture Notes in Mathematics, 728 (1978 · Zbl 0982.53033
[13] [13] , Notes on hyperbolic groups, Group Theory from a Geometrical Viewpoint, World Scientific, 1990, 3-63. · Zbl 0849.20023
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