Geometry & Topology Seminar: Pappus's Theorem, Patterns of Geodesics, and the Barbot Component

Abstract: Pappus's Theorem is a classic theorem in projective geometry. First, I will explain how the iteration of Pappus's Theorem leads to representations of the modular group into Isom(X), where X is the 5-dimensional symmetric space associated to SL_3(R). Second, I will explain how the famous Farey triangulation in the hyperbolic plane can be placed inside X and then bent like a pleated plane. The Pappus groups are symmetry groups of these high rank pleated planes. Third, if time permits, I will say a few words about how this perspective combines with some computer algebra to give a complete characterization of one of the components of the representation variety of discrete faithful representations of the modular group into Isom(X), the so-called Barbot component.

Host: Steven Frankel