Pappus Theorem, Schwartz Representations and Anosov Representations

TL;DR

This paper explores the relationship between Schwartz representations derived from Pappus's theorem and Anosov representations, showing Schwartz representations as limits of certain Anosov representations of the modular group.

Contribution

It constructs a 2-dimensional family of Anosov representations extending Schwartz representations, linking them as limits of Anosov representations into the projective symmetry group.

Findings

Schwartz representations are limits of Anosov representations.

A 2-dimensional family of Anosov representations is constructed.

Some representations extend from subgroup to the entire group.

Abstract

In the paper "Pappus's theorem and the modular group", R. Schwartz constructed a 2-dimensional family of faithful representations $\rho_\Theta$ of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ into the group $\mathscr{G}$ of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup $\mathrm{PSL}(2,\mathbb{Z})_o$ of $\mathrm{PSL}(2,\mathbb{Z})$ under each representation $\rho_\Theta$ is in the subgroup $\mathrm{PGL}(3,\mathbb{R})$ of $\mathscr{G}$ and preserves a topological circle in the flag variety, but $\rho_\Theta$ is not Anosov. In her PhD Thesis, V. P. Val\'erio elucidated the Anosov-like feature of Schwartz representations: For every $\rho_\Theta$, there exists a 1-dimensional family of Anosov representations $\rho^\varepsilon_{\Theta}$ of $\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ whose limit is the restriction of $\rho_\Theta$ to $\mathrm{PSL}(2,\mathbb{Z})_o$. In this paper, we improve her work: For each $\rho_\Theta$, we build a 2-dimensional family of Anosov representations of $\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ containing $\rho^\varepsilon_{\Theta}$ and a 1-dimensional subfamily of which can extend to representations of $\mathrm{PSL}(2,\mathbb{Z})$ into $\mathscr{G}$. Schwartz representations are therefore, in a sense, the limits of Anosov representations of $\mathrm{PSL}(2,\mathbb{Z})$ into $\mathscr{G}$.