On Pappus and Anosov Representations of the Modular Group

Abstract

Let $X=S{L}_3(\mathbb{R})/SO(3)$. Let $\mathcal{D}\mathcal{F}\mathcal{R}$ be the space of discrete faithful representations of the modular group into ${\rm Isom\/}(X)$ 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 ${\mathbb{R}}^2\times [0,\infty )$. The boundary of $\mathcal{B}$ parametrizes the Pappus representations and the interior consists of Anosov representations.

This is an edited and improved subset of a very long paper of mine, arXiv:2412.18547. In this shorter paper I isolate one of the main results and give a self-contained proof. I also take the opportunity to fix a few glitches and add some helpful details. Computer assisted - Mathematica files downloadable from my website