# Homogeneity in the free group

@article{Perin2010HomogeneityIT, title={Homogeneity in the free group}, author={Chlo{\'e} Perin and Rizos Sklinos}, journal={arXiv: Group Theory}, year={2010} }

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.

## 38 Citations

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