Patterns of Geodesics, shearing, and Anosov representations of the modular group

This talk covers both old and new work of mine. I'll first explain how you can use Pappus's Theorem to get relatively Anosov representations of the modular group into Isom(X), where X is the symmetric space $S{L}_3(R)/SO(3)$. 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 edges of the Farey triangulation but are bent like a pleated planes. Finally, I will at least say a few words about how this picture enhances the work of Barbot, Lee, and Valerio on Anosov representations of the modular group into Isom(X) and leads to a complete classification of the Barbot component of such representations.