Geometric Group Theory in 3-Manifolds

Presenter: Bryce Iversen

Presenter Status: Undergraduate student

Academic Year: 22-23

Semester: Spring

Faculty Mentor: Ben Ford

Department: Mathematics

Funding Source/Sponsor: McNair

President's Strategic Plan Goal: Connectivity and Community Engagement

Abstract:
Geometric group theory is a field of mathematics centered around understanding the properties of algebraic structures called groups through geometry. Geometric group theory can be applied to understand specific types of mathematical objects called 3-manifolds. A manifold is a topological space that, within a small enough region, resembles some Euclidean space. A topological space is a way of redefining space using selected properties. For example, you might choose to describe space in the way that most people find natural: the Euclidean metric. But, you could choose to describe space using properties other than distance. A 2-manifold represents the surface of a 2-dimensional mathematical object. For example, the surface of the Earth is a 2-manifold. A human, despite living in a sphere, would see themselves as living on a plane since the radius of Earth is large enough that something as small as a human is unable to view the totality of Earth. Equivalently, our universe can be thought of as a 3-manifold. To a small enough observer, say a human looking out into the stars, our universe appears to be flat Euclidean 3-space. However, if we assume the universe is a 3-manifold, it could look very different to a larger observer. Our research centers around the implications of geometric group theory on 3-manifolds. We explore the implications of the following questions. How does geometric group theory connect group theory and manifolds? What are some examples of 3-manifolds that satisfy the Poincaré duality theorem? How can geometric group theory help us visualize these 3-manifolds that do satisfy the Poincaré duality theorem? How does geometric group theory help us to describe Coxeter groups? What are some specific examples of Coxeter groups?