## Free monoids in hyperbolic groups. (Monoïdes libres dans les groupes hyperboliques.)(French)Zbl 0973.20036

Séminaire de théorie spectrale et géométrie. Année 1999-2000. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 18, 157-170 (2000).
The authors prove the following Theorem: Let $$\Gamma$$ be a $$\delta$$-hyperbolic group equipped with a given finite system of generators. Then there exists an explicit constant $$n$$ such that for any elements $$f$$ and $$g$$ of $$\Gamma$$, there exists $$\varepsilon=\pm 1$$ such that $$f^n$$ and $$g^{\varepsilon n}$$ are nontrivial, and such that either the group generated by $$f^n$$ and $$g^{\varepsilon n}$$ is $$\mathbb{Z}$$, or these two elements freely generate as monoid (or semi-group) a free monoid.
For the entire collection see [Zbl 0955.00015].

### MSC:

 20F67 Hyperbolic groups and nonpositively curved groups 20F05 Generators, relations, and presentations of groups 20M05 Free semigroups, generators and relations, word problems
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