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

Champetier, Christophe; Guirardel, Vincent

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].

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

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

Keywords:

word hyperbolic groups; finitely generated groups; free monoids