Date of Graduation

5-2024

Document Type

Thesis

Degree Name

Bachelor of Science in Mathematics

Degree Level

Undergraduate

Department

Mathematical Sciences

Advisor/Mentor

Day, Matthew

Committee Member/Reader

Hare, Laurence

Committee Member/Second Reader

Vyas, Reeta

Committee Member/Third Reader

Miller, Lance

Abstract

Surfaces have long been a topic of interest for scholars inside and outside of mathe- matics. In a topological sense, surfaces are spaces which appear flat on a local scale. Surfaces in this sense have a restricted set of properties, including the behavior of loops around a surface, codified in the fundamental group.

All but 3 surface groups have been shown to embed into a class of groups called right-angled Artin groups. The method through which these embeddings are created places large restrictions on all homomorphisms from surface groups to right-angled Artin groups.

One such restriction on these homomorphisms is on distortion. A group can be represented by a graph, where distortion codifies how shortcuts can be found outside of a subgraph. It has already been shown that, for all relevant surface groups, there exists at least one undistorted embedding inside of a right-angled Artin group. Theorem A and Conjecture B seek to extend this to all homomorphisms.

Keywords

Distortion, Surfaces, Surface Groups, Graph Groups, Right-Angled Artin Groups

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