Available at: https://digitalcommons.calpoly.edu/theses/2756
Date of Award
6-2023
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 group $\Gamma$ with presentation $\relgroup{\scr{\scr{A}}}{\scr{R}}$, a natural question, known as the word problem, is how does one decide whether or not two words in the free group, $F(\scr{\scr{A}})$, represent the same element in $\Gamma$. In this thesis, we study certain aspects of geometric group theory, especially ideas published by Gromov in the late 1980's. We show there exists a quasi-isometry between the group equipped with the word metric, and the space it acts on. Then, we develop the notion of a CAT(0) space and study groups which act properly and cocompactly by isometries on these spaces, such groups are known as CAT(0) groups. Furthermore, we show CAT(0) groups have a solvable word problem.