Representations of the 2-Torus in the 3-Torus

Student: Bryce Iversen

Faculty Mentor: Ben Ford

Mathematics and Statistics

College of Science, Technology, and Business

Geometric group theory can be applied to infer the structure of certain groups using specific types of geometric mathematical objects called manifolds. A manifold is a space that, within a small enough region, resembles some Euclidean space. We study a particular 3-manifold called the 3-torus—which locally resembles R³—from the perspective of geometric group theory. It is known that simple closed curves (non-intersecting loops) on the surface of the 2-torus (a doughnut) imply a relationship between SL(2, Z) acting on the Farey Graph and the Mapping Class Group of the 2-torus acting on the Curve Graph of the 2-torus. Indeed, SL(2, Z) is isomorphic to the Mapping Class Group of the 2-torus and, similarly, the Farey Graph is isomorphic to the Curve Graph. By moving one dimension higher, we aim to geometrically prove a known result that the Mapping Class Group of the 3-torus is isomorphic to SL(3, Z). To this end, we use computer graphics simulations to visualize representations of the 2-torus in the 3-torus, which we hypothesize are analogues to simple closed curves on the 2-torus in the two-dimensional case.