Pappus's Theorem, Patterns of Geodesics, and Representations of the Modular Group

by Richard Schwartz (Brown University)

Monday Jun 16, 2025, 2:00 PM → 3:15 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

This talk is about a mixture of old and new work. First I will talk about how you can iterate Pappus's theorem and construct a 2-parameter family of relatively Anosov representations of the modular group into Isom(X), where X = SL(3,R)/SO(3). Then I will explain how to interpret these representations as symmetry groups of patterns of geodesics in X that have the same asymptotic properties as the Farey graph in the hyperbolic plane. Finally I will say a few words about how this picture allows for a complete classification of the Barbot component of discrete faithful representations of the modular group into Isom(X).