Homogeneity in the free group

Abstract

We show that any non abelian free group $\F$ is strongly ${\aleph }_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $\Aut(\F)$. We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not ${\aleph }_0$-homogeneous.

26 pages

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Cited by in corpus (11)

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• Hyperbolic towers and independent generic sets in the theory of free groups
• Ampleness in the free group
• Homogeneity and prime models in torsion-free hyperbolic groups
• Algebraic & definable closure in free groups
• On first order rigidity for linear groups
• Forking and JSJ decompositions in the free group II
• Towers and the first-order theory of hyperbolic groups
• Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry
• Maximal hyperbolic towers and weight in the theory of free groups
• Non-$\forall$-homogeneity in free groups