Representations of the modular group into the isometries of SL(3, R)/SO(3)

TL;DR

This paper explores a specific connected component of the space of representations of the modular group into the isometry group of a symmetric space, identifying which are Anosov and relating to prior constructions by Schwartz.

Contribution

It characterizes a connected component of the representation space, including Schwartz's representations and their Anosov deformations, providing new insights with different proofs.

Findings

Certain representations are proven to be Anosov.

The connected component includes Schwartz's representations.

New proofs of previously known results are provided.

Abstract

We describe a connected component of the space of conjugacy classes of representations of the modular group $\mathrm{PSL}_2(\mathbb{Z})$ into the isometry group of the symmetric space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$. This connected component contains the family of representations constructed by Schwartz via Pappus' theorem, as well as their Anosov deformations studied by Barbot, Lee, and Val\'erio. We show that certain representations in this component (far from the Schwartz representations) are Anosov. The main results of this paper were previously proved by Schwartz in arXiv:2412.18457, though with different arguments. I was unaware of arXiv:2412.18457 when I posted the first version of this paper.