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.

Nepsa Thesis.pdf (2526 kB)

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