Dissertations, Theses, and Capstone Projects

Date of Degree

6-2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Mathematics

Advisor

Vladimir Shpilrain

Committee Members

Ilya Kapovich

Alina Vdovina

Subject Categories

Other Mathematics

Keywords

group theory, combinatorial group theory, free groups

Abstract

One of the fundamental problems in the field of combinatorial group theory is telling when two group presentations represent isomorphic groups. Since applying a free group automorphism to the set of relators of a presentation gives an isomorphic group, we want to be able to quickly decide when looking at a relator whether a given word can be sent to it via an automorphism. We give a hands-on introduction to the automorphisms of free groups using patterns of colored beads. We show that you can make this decision correctly in constant time on average by looking for "orbit-blocking" words that guarantee the answer is no if they appear as subwords of the input word.

Since we know some nice, isomorphism-invariant properties of one-relator groups where the relator is positive (that is, only written in positive generators), we'll also address the problem of recognizing words that can be sent to positive words via an automorphism. We call such words potentially positive. We will say what such words look like in the free group of rank two, then demonstrate a new fastest known algorithm for recognizing them. Finally, we discuss how common or uncommon it is for a word to be potentially positive, and address the language class complexity of potential positivity.

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