DRP – Fall 2021

A central theme of geometric group theory is to understand finitely generated groups by studying spaces on which they act. For example, particularly nice actions on trees illuminate the structures of certain groups. One important group is $\mathsf{SL_2(\Z)}$ , the group of $\mathsf{2 \times 2}$ integer matrices with determinant $\mathsf{1}$. We will investigate the general theory of groups acting on trees, and use these results to ultimately find a presentation of $\mathsf{SL_2(\Z)}$ by considering its natural action on the Farey tree.

The field of $\mathsf{p}$-adic numbers is a completion of the rational numbers $\Q$. This presentation will introduce the $\mathsf{p}$-adic valuations and $\mathsf{p}$-adic expansions. We will cover fundamental aspects of the $\mathsf{p}$-adic numbers, including an overview of their derivation and their applications in algebra, analysis, and topology. A basic understanding of modular arithmetic, real analysis, group theory, and ring theory is presumed.

We start with talking about the ZF axioms, and then we discuss the construction of the natural numbers, integers, and rational numbers briefly. Then we sketch the construction of the real numbers using the notion of Dedekind cuts.

In this presentation, we will define knots in the mathematical sense and explore methods of distinguishing knots through the use of knot invariants. In particular, we will discuss knot quandles and how they can be used to generate these invariants.