Classifying Groups Up to Quasi-Isometry Using Cayley Graphs
The extensive study of maps between groups has led mathematicians to develop several theorems regarding the classification of groups up to isomorphism. A less studied field is that of quasi-isometries between groups, which are functions that allow groups to differ structurally while maintaining an "almost" isomorphic relationship. The freedom permitted by the less restrictive quasi-isometries makes more difficult the task of classifying groups up to quasi-isometry. The search for quasi-isometry invariants and classes of groups that are quasi-isometrically rigid are the two main areas of the classification. In this paper, we will prove some properties of quasi-isometries and investigate quasi-isometric relationships between the groups ZxZ, Z*Z, and Z2*Z3. Cayley graphs for these groups, as well as similarities in their structures, will be the main discovery tools. Exploring these proofs and examples will assist in the future study of the quasi-isometries discovered between classes of groups such as the Baumslag-Solitar groups. In the end, I hope to have offered some insight into this topic in such as way that non-mathematicians and mathematicians alike will understand and appreciate this beautiful concept.
School:
Truman State University
Department:
Mathematics
Research Advisor:
Kim Whittlesey
Department of Research Advisor:
Mathematics
Year of Publication:
2003