We show that any nonabelian free group is strongly -homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under . 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 -homogeneous.
Chloé Perin. Rizos Sklinos. "Homogeneity in the free group." Duke Math. J. 161 (13) 2635 - 2668, 1 October 2012. https://doi.org/10.1215/00127094-1813068