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

TL;DR

This work characterizes the connected component $X_0(\mathrm{PSL}_2(\mathbb{Z}),\mathrm{Isom}(X))$ of representations into the isometry group of the symmetric space $X=\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$ that contains Schwartz’s Schwartz space, showing it is homeomorphic to $\mathbb{R}^3$ and describing its quotient model as $(S^1\times\mathbb{Z}_2)\backslash(\mathbb{H}^2\times\mathbb{H}^2)$. It identifies Fuchsian loci of type I and II as a proper curve, with the union of these loci corresponding to representations preserving totally geodesic $\mathbb{H}^2$ inside $X$, and it provides a coordinate system $(s,t,\vartheta)$ for the character variety, where $s=0$ (resp. $t=0$) corresponds to type I (resp. II) Fuchsian representations. The paper also analyzes the Schwartz space via a peripheral trace condition and proves that certain non-Schwartz representations in the component are Anosov by using asymptotic and Morse-theoretic arguments on $X$, including triangle approximations in the parallel set $P(\gamma)$ and the Morse characterization of Anosov groups. Overall, the results connect the geometry of $X$ with the dynamics of PSL$_2(\mathbb{Z})$ representations, enriching the understanding of Anosov deformations in higher-rank symmetric spaces.

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 Valério. 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.

Figures (10)

• Figure 1: The plane ${\lambda_1+\lambda_2+\lambda_3=0}$ with the three singular lines through the origin.
• Figure 2: The spherical Weyl chamber $\sigma\subset\partial_\infty F$ and the Weyl sector $V(x,\sigma)\subset F$, in the same picture of a maximal flat $F$ as in Figure \ref{['Figure:MaxFlat']}. In the plane $\lambda_1+\lambda_2+\lambda_3=0$, $V(x,\sigma)$ is defined by $\lambda_1\geq\lambda_2\geq \lambda_3$.
• Figure 3: The triangle $\bar{x}\bar{y}\bar{z}$ in the plane $\{ h_0\}\times\mathbb H^2$.
• Figure 4: The $\zeta$-angle $\angle^\zeta_p(q,q')=\angle_p(r,r')$.
• Figure 5: The maximal flat $F(\sigma_-,\sigma_+)$ as in Theorem \ref{['Thm:LocalMorse']}

Theorems & Definitions (74)

• Theorem 1.1: Schwartz2025
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4: Schwartz2025
• Theorem 1.5
• Remark 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• Definition 2.4