Homogeneity in the free group

Details Homogeneity in the free group Homogeneity in the free group

2010-03-22

arxiv.org/abs/1003.4095v1

Mathematics

Group Theory

26 pages

Scientific paper

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.

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