Elementary embeddings in torsion-free hyperbolic groups.
(Plongements élémentaires dans des groupes hyperboliques sans torsion.)

*(English. French summary)*Zbl 1245.20052
Ann. Sci. Éc. Norm. Supér. (4) 44, No. 4, 631-681 (2011); erratum ibid. 46, No. 5, 851-856 (2013).

The author uses a combination of geometric methods and of methods from mathematical logic to obtain results on combinatorial group theory. This combination of methods is now classical, and the author is inspired by techniques introduced by Z. Sela in a series of papers that were published in the last decade, which culminated in a positive answer to a problem of Tarski asking whether any two finitely generated non-Abelian free groups are elementary equivalent (a result also obtained by Kharlampovich and Myasnikov).

The main result of the paper under review is the following Theorem: Let \(G\) be a torsion-free hyperbolic group and let \(H\hookrightarrow G\) be an elementary embedding. Then \(G\) is a hyperbolic tower based on \(H\).

Here, a subgroup \(H\) of a group \(G\) is said to be elementary if any tuple of elements of \(H\) satisfies the same first-order properties in \(H\) and \(G\). Roughly speaking, a hyperbolic tower based on a group \(H\) is a group obtained by successive addition of hyperbolic floors to a “ground floor” which is the free product of \(H\), closed surface groups and a free group. The notion is due to Sela.

In the special case where \(G\) is a free group, a consequence of the above theorem is the following: Theorem: Let \(H\) be a proper subgroup of the free group \(\mathbb F_n\). Then the embedding of \(H\) in \(\mathbb F_n\) is elementary if and only if \(H\) is a non-Abelian free factor of \(\mathbb F_n\).

The author also proves the following Theorem: Let \(S\) be the fundamental group of a closed hyperbolic surface \(F\) and let \(H\) be a proper non-elementary subgroup of \(S\). Then \(H\) is a non-Abelian free factor of the fundamental group of a connected surface \(\Sigma_0\) of \(\Sigma\) whose complement in \(\Sigma\) is connected and which satisfies \(|\chi(\Sigma_0)|\leq|\chi(\Sigma)|/2\), with equality if and only if \(\Sigma\) is the double of \(\Sigma_0\).

The main result of the paper under review is the following Theorem: Let \(G\) be a torsion-free hyperbolic group and let \(H\hookrightarrow G\) be an elementary embedding. Then \(G\) is a hyperbolic tower based on \(H\).

Here, a subgroup \(H\) of a group \(G\) is said to be elementary if any tuple of elements of \(H\) satisfies the same first-order properties in \(H\) and \(G\). Roughly speaking, a hyperbolic tower based on a group \(H\) is a group obtained by successive addition of hyperbolic floors to a “ground floor” which is the free product of \(H\), closed surface groups and a free group. The notion is due to Sela.

In the special case where \(G\) is a free group, a consequence of the above theorem is the following: Theorem: Let \(H\) be a proper subgroup of the free group \(\mathbb F_n\). Then the embedding of \(H\) in \(\mathbb F_n\) is elementary if and only if \(H\) is a non-Abelian free factor of \(\mathbb F_n\).

The author also proves the following Theorem: Let \(S\) be the fundamental group of a closed hyperbolic surface \(F\) and let \(H\) be a proper non-elementary subgroup of \(S\). Then \(H\) is a non-Abelian free factor of the fundamental group of a connected surface \(\Sigma_0\) of \(\Sigma\) whose complement in \(\Sigma\) is connected and which satisfies \(|\chi(\Sigma_0)|\leq|\chi(\Sigma)|/2\), with equality if and only if \(\Sigma\) is the double of \(\Sigma_0\).

Reviewer: Athanase Papadopoulos (Strasbourg)

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20A15 | Applications of logic to group theory |

03C60 | Model-theoretic algebra |

20E05 | Free nonabelian groups |

03C07 | Basic properties of first-order languages and structures |