## Duke Mathematical Journal

### Homogeneity in the free group

#### Abstract

We show that any nonabelian free group $\mathbb {F}$ is strongly $\aleph_{0}$-homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $\operatorname {Aut}(\mathbb {F})$. 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 strongly $\aleph_{0}$-homogeneous.

#### Article information

Source
Duke Math. J., Volume 161, Number 13 (2012), 2635-2668.

Dates
First available in Project Euclid: 11 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1349960279

Digital Object Identifier
doi:10.1215/00127094-1813068

Mathematical Reviews number (MathSciNet)
MR2988905

Zentralblatt MATH identifier
1270.20028

#### Citation

Perin, Chloé; Sklinos, Rizos. Homogeneity in the free group. Duke Math. J. 161 (2012), no. 13, 2635--2668. doi:10.1215/00127094-1813068. https://projecteuclid.org/euclid.dmj/1349960279

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