Homogeneity and prime models in torsion-free hyperbolic groups

Abderezak Ould Houcine

Homogeneity and prime models in torsion-free hyperbolic groups

Group Theory 2010-04-28 v1 Logic

Authors: Abderezak Ould Houcine

Abstract

We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $\bar a$ , $\bar b \in F^n$ , having the same complete $n$ -type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$ . We further study existential types and we show that for any tuples $\bar a, \bar b \in F^n$ , if $\bar a$ and $\bar b$ have the same existential $n$ -type, then either $\bar a$ has the same existential type as a power of a primitive element, or there exists an existentially closed subgroup $E(\bar a)$ (resp. $E(\bar b)$ ) of $F$ containing $\bar a$ (resp. $\bar b$ ) and an isomorphism $\sigma : E(\bar a) \to E(\bar b)$ with $\sigma(\bar a)=\bar b$ . We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are $\exists$ -homogeneous and prime. This gives, in particular, concrete examples of finitely generated groups which are prime and not QFA.

Keywords

group theory hyperbolic groups finite group theory

Cite

@article{arxiv.1004.4698,
  title  = {Homogeneity and prime models in torsion-free hyperbolic groups},
  author = {Abderezak Ould Houcine},
  journal= {arXiv preprint arXiv:1004.4698},
  year   = {2010}
}

Homogeneity and prime models in torsion-free hyperbolic groups

Abstract

Keywords

Cite

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