On Pappus and Anosov Representations of the Modular Group

Let $X=S{L}_3(\mathbb{R})\mathrm{/}SO(3)$. Let $\mathcal{D}\mathcal{F}\mathcal{R}$ be the space of discrete faithful representations of the modular group into $\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}\text{/}(X)$ which map the order $2$ generator to an isometry with a unique fixed point. In this paper, we prove that $\mathcal{D}\mathcal{F}\mathcal{R}$ has a component $\mathcal{B}$, the so-called Barbot component, that is homeomorphic to ${\mathbb{R}}^2\times [0,\mathrm{\infty })$. The boundary of $\mathcal{B}$ parametrizes the Pappus representations and the interior consists of Anosov representations.