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. 

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References

SHOWING 1-10 OF 33 REFERENCES

Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free

On the generic type of the free group

  • R. Sklinos
  • Mathematics
    The Journal of Symbolic Logic
  • 2011
TLDR
It is proved that the set of primitive elements in finite rank free groups is not uniformly definable and that uncountable free groups are not ℵ1-homogeneous.

Diophantine geometry over groups VI: the elementary theory of a free group

Abstract.This paper is the sixth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined

Trees of cylinders and canonical splittings

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively

Aspects of free groups

Elementary embeddings in torsion-free hyperbolic groups

We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We

Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II

Abstract. We borrow the Jaco-Shalen-Johannson notion of characteristic sub-manifold from 3-dimensional topology to study cyclic splittings of torsion-free (Gromov) hyperbolic groups and finitely

Actions of finitely generated groups on R-trees

We study actions of finitely generated groups on $\bbR$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or

JSJ-Decompositions of finitely presented groups and complexes of groups

Abstract.A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN

Cut points and canonical splittings of hyperbolic groups

In this paper, we give a construction of the JSJ splitting of a one-ended hyperbolic group (in the sense of Gromov [Gr]), using the local cut point structure of the boundary. In particular, this