#### Date of Award

Fall 2021

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Edward K. Hinson

#### Second Advisor

Maria Basterra

#### Third Advisor

Adam Boucher

#### Abstract

For any commutative ring R with identity, Um(2, R) is the set of all vectors α ∈ R^2 such that α^T β = 1 for some β ∈ R^2. Motivated by Hinson and Samuel, we endow Um(2, R) with a pseudo-graph structure and introduce a family of functions on the general ring R defined in terms of graph theoretic distance from a designated base-point or base-set. We propose a particular connected base-set whose quasi-Euclidean function exhibits the most computationally convenient properties. Unlike in Um(n, R), n ≥ 3, the relationship between path components of Um(2, R) and its orbits under elementary matrix action is complicated, and we develop tools to analyze this case. The main such tool uses closed paths in Um(2, R) satisfying certain properties with respect to actions of elementary orthogonal matrices. Among our applications are: the equivalence of path-connectedness of Um(2, R) and the GE2 status of R; recovering Cohn’s result that for F a field, SL(2, F[x, y]) ̸= E(2, F[x, y]);and demonstrating Um(2, F[x, y]) has infinitely many distinct path components.

#### Recommended Citation

Fill, Timothy A., "Path Components and Elementary Orbits of Um(2,R)" (2021). *Doctoral Dissertations*. 2615.

https://scholars.unh.edu/dissertation/2615