#### 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).

#### Recommended Citation

Natoli, Christopher James, "Some Model Theory of Free Groups" (2021). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/4132