Non--homogeneity in free groups
arXiv:2001.09904
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.