Van Kampen Diagrams and Small Cancellation Theory

Author

Kelsey N. Lowrey, California Polytechnic State University, San Luis ObispoFollow

DOI: https://doi.org/10.15368/theses.2022.37
Available at: https://digitalcommons.calpoly.edu/theses/2647

Date of Award

6-2022

Degree Name

MS in Mathematics

Department/Program

Mathematics

College

College of Science and Mathematics

Advisor

Anton Kaul

Advisor Department

Mathematics

Advisor College

College of Science and Mathematics

Abstract

Given a presentation of G, the word problem asks whether there exists an algorithm to determine which words in the free group, F(A), represent the identity in G. In this thesis, we study small cancellation theory, developed by Lyndon, Schupp, and Greendlinger in the mid-1960s, which contributed to the resurgence of geometric group theory. We investigate the connection between Van Kampen diagrams and the small cancellation hypotheses. Groups that have a presentation satisfying the small cancellation hypotheses C'(1/6), or C'(1/4) and T(4) have a nice solution to the word problem known as Dehn’s Algorithm.