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 $\rho_\Theta$ 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 $\mathscr{G}$.