Pappus Theorem, Schwartz Representations and Anosov Representations

Abstract

In the paper "Pappus's theorem and the modular group", R. Schwartz constructed a 2-dimensional family of faithful representations $ρ_Θ$ 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 $ρ_Θ$ is in the subgroup $\mathrm{PGL}(3,\mathbb{R})$ of $\mathscr{G}$ and preserves a topological circle in the flag variety, but $ρ_Θ$ is not Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: For every $ρ_Θ$, there exists a 1-dimensional family of Anosov representations $ρ^\varepsilon_Θ$ of $\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ whose limit is the restriction of $ρ_Θ$ to $\mathrm{PSL}(2,\mathbb{Z})_o$. In this paper, we improve her work: For each $ρ_Θ$, we build a 2-dimensional family of Anosov representations of $\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ containing $ρ^\varepsilon_Θ$ 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 $\mathscr{G}$.