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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 \(\leq M_G\).

MSC:

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
57M07 Topological methods in group theory
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References:

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