Submission date: 24 March 2010.

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.

Mathematics Subject Classification: 20E05, 20F67, 03C07

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