Date of Degree

2-2021

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Olga Kharlampovich

Committee Members

Ilya Kapovich

Vladimir Shpilrain

Subject Categories

Algebra | Logic and Foundations

Keywords

group theory, model theory, free groups

Abstract

There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.

The second main result is that if a countable group is elementarily equivalent to a non-abelian free group and all of its finitely generated abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

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