Le Retour de Pappus

TL;DR

This paper connects Pappus's Theorem to geometric representations of the modular group, realizing these as isometry groups of geodesic patterns in a symmetric space, and explores their deformation and bending phenomena.

Contribution

It constructs a 2-parameter family of geometric representations of the modular group as isometries of geodesic patterns in a symmetric space, extending previous algebraic work.

Findings

Realization of modular group representations as isometry groups

Deformation of Farey triangulation patterns in symmetric space

Identification of a bending phenomenon in geodesic patterns

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.