Submission date: 27 January 2020

Abstract:

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not ∀-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fraïssé classes in which embeddings must preserve ∀-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.

Mathematics Subject Classification: 03C60 (Primary) 20E05, 20F70 (Secondary)

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Full text arXiv 2001.09904: pdf, ps.