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**Homogeneity in relatively free groups.**
*(English)*
Zbl 1276.20030

The article contributes to the study of free groups, a topic which has been of the greatest interest to model theorists since Sela’s work bringing together geometric group theory and model theory. The main concept of the article, homogeneity, is borrowed from model theory; it describes the possibility of extending so-called “elementary” (that is, with nice logical properties) partial monomorphisms from a group into itself. The article is however entirely group-theoretic in nature and the non-logician will find it to be self-contained; this includes a precise definition of homogeneity.

From the introduction: “A. Ould Houcine [Confluentes Math. 3, No. 1, 121-155 (2011; Zbl 1229.20020)] and independently C. Perin and R. Sklinos [Duke Math. J. 161, No. 13, 2635-2668 (2012; Zbl 1270.20028)] proved that any free group of finite rank is strongly homogeneous. […] R. Sklinos [J. Symb. Log. 76, No. 1, 227-234 (2011; Zbl 1213.03047)] showed that any free group of uncountable rank is not \(\aleph_1\)-homogeneous. His proof was based on the deep Sela’s result on stability of the theory of free groups and used some sophisticated technique of model-theoretic stability theory. The aim of this note is to give a simple direct proof of a more general result: any torsion-free, residually finite relatively free group \(F\) of infinite rank is not \(\aleph_1\)-homogeneous.”

As a matter of fact, the proof is completely elementary: an easily constructed elementary map is proved to refute \(\aleph_1\)-homogeneity by quick, elegant, and classical group-theoretic means.

From the introduction: “A. Ould Houcine [Confluentes Math. 3, No. 1, 121-155 (2011; Zbl 1229.20020)] and independently C. Perin and R. Sklinos [Duke Math. J. 161, No. 13, 2635-2668 (2012; Zbl 1270.20028)] proved that any free group of finite rank is strongly homogeneous. […] R. Sklinos [J. Symb. Log. 76, No. 1, 227-234 (2011; Zbl 1213.03047)] showed that any free group of uncountable rank is not \(\aleph_1\)-homogeneous. His proof was based on the deep Sela’s result on stability of the theory of free groups and used some sophisticated technique of model-theoretic stability theory. The aim of this note is to give a simple direct proof of a more general result: any torsion-free, residually finite relatively free group \(F\) of infinite rank is not \(\aleph_1\)-homogeneous.”

As a matter of fact, the proof is completely elementary: an easily constructed elementary map is proved to refute \(\aleph_1\)-homogeneity by quick, elegant, and classical group-theoretic means.

Reviewer: Adrien Deloro (Paris)

### MSC:

20E10 | Quasivarieties and varieties of groups |

03C50 | Models with special properties (saturated, rigid, etc.) |

03C60 | Model-theoretic algebra |

20E05 | Free nonabelian groups |

20A15 | Applications of logic to group theory |

20E26 | Residual properties and generalizations; residually finite groups |

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

### Keywords:

varieties of groups; relatively free groups; homogeneous structures; residually finite groups; homogeneity; stable groups; elementary monomorphisms
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\textit{O. Belegradek}, Arch. Math. Logic 51, No. 7--8, 781--787 (2012; Zbl 1276.20030)

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### References:

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[9] | Perin, C., Sklinos, R.: Homogeneity in the free group. ArXiv–Mathematics http://arxiv.org/pdf/1003.4095.pdf . 22 March 2010 · Zbl 1270.20028 |

[10] | Sela Z.: Diophantine geometry over groups. VI. The elementary theory of a free group. Geom. Funct. Anal. 16, 707–730 (2006) · Zbl 1118.20035 |

[11] | Sela, Z.: Diophantine geometry over groups. VIII. Stability. ArXiv–Mathematics http://arxiv.org/pdf/math/0609096v1.pdf . 4 Sept 2006 · Zbl 1285.20042 |

[12] | Sklinos R.: On the generic type of the free group. J. Symb. Log. 76, 227–234 (2011) · Zbl 1213.03047 |

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