| Résume | The Tarski problem asks whether all finitely generated non-abelian free groups share the same first-order theory. In 2006, Z. Sela answered this question affirmatively. Building on this result, a natural extension is to explore which groups can or cannot be distinguished from non-abelian free groups by first-order sentences. We prove that almost all the (finitely generated) groups cannot be distinguished from non-abelian free groups by a given first-order sentence. Namely, we prove that for a random group (in the Gromov density model, for any density d<0.5), the group cannot be distinguished from a finitely generated non-abelian free group by a given (minimal rank) sentence, in overwhelming probability. |
