Le Retour de Pappus

TL;DR

This work realizes the Pappus modular groups as isometry groups of Farey-patterns embedded in the symmetric space $X=SL_3(\mathbb{R})/SO(3)$, extending Schwartz's earlier 2-parameter moduli of representations into a geometric framework. By encoding Pappus configurations with convex marked boxes and dualities, the author constructs a canonical Farey pattern $\Gamma_{\mathcal M}$ whose geodesics correspond to the marked-box data and whose modular actions intertwine combinatorial and geometric symmetries. A key contribution is the introduction of the triple invariant and the prism/bending picture, which yields iso-prismatic families of patterns with a controlled 1-parameter bending between adjacent prisms and a pleated-surface filling that captures the asymptotic structure of the patterns. This approach connects the Pappus modular groups to higher Teichmüller theory through relatively Anosov representations, enriching the understanding of deformations, limit sets, and symmetry in higher rank symmetric spaces.

Abstract

In my 1993 paper, "Pappus's Theorem and the Modular Group", I explained how the iteration of Pappus's Theorem gives rise to a $2$-parameter family of representations of the modular group into the group of projective automorphisms. In this paper we realize these representations as isometry groups of patterns of geodesics in the symmetric space $X=SL_3(\R)/SO(3)$. The patterns have the same asymptotic structure as the geodesics in the Farey triangulation, so our construction gives a $2$ parameter family of deformations of the Farey triangulation inside $X$. We also describe a bending phenomenon associated to these patterns.

Theorems & Definitions (19)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Lemma 3.2: Compatibility
• Lemma 3.3
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4