Elementary embeddings in torsion-free hyperbolic groups

Perin, Chloé

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 4, pp. 631-681.

On obtient une description des plongements élémentaires (au sens de la logique du premier ordre) dans un groupe hyperbolique sans torsion, en termes de tours hyperboliques de Sela. Ainsi, si $H$ est plongé élémentairement dans un groupe hyperbolique sans torsion $\Gamma$, on peut obtenir $\Gamma$ en amalgamant successivement des groupes de surfaces à bord à un produit libre de $H$ avec des groupes libres et des groupes de surfaces fermées. Ceci permet en corollaire de montrer qu’un sous-groupe plongé élémentairement dans un groupe libre de type fini est un facteur libre. On considère également le cas où $\Gamma$ est le groupe fondamental d’une surface hyperbolique fermée. Les techniques utilisées pour obtenir cette description sont essentiellement géométriques : actions sur des arbres réels ou simpliciaux, décompositions JSJ. On s’appuie également sur des résultats d’existence d’ensembles de factorisation utilisés dans la construction de diagrammes de Makanin-Razborov pour un groupe hyperbolique sans torsion.

We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma$, we show that the group $\Gamma$ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the special case where $\Gamma$ is the fundamental groups of a closed hyperbolic surface. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, a result used in the construction of Makanin-Razborov diagrams for torsion-free hyperbolic groups.

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