Non-∀-homogeneity in free groups

2020/01/27 by Olga Kharlampovich, Kharlampovich, Olga, Christopher Natoli +1
Mathematics · #Advanced Topology and Set Theory #Geometric and Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/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.

Related

• The construction principle and non homogeneity of uncountable relatively free groups
• Virtually free groups are almost homogeneous
• Simple groups separated by finiteness properties
• Hereditarily separable groups and monochromatic uniformization
• Non-hopfian relatively free groups