Entropy Rigidity for Maximal Representations

TL;DR

The paper proves entropy rigidity for maximal representations $\rho:\Gamma\to \mathsf{Sp}(2n,\mathbb{R})$ with $\Gamma$ a lattice in $\mathsf{PSL}(2,\mathbb{R})$, by developing a measurable $(1,1,2)$-hypertransversality for the $\rho$-limit map and constructing Patterson–Sullivan and Bowen–Margulis–Sullivan measures. It shows the renormalized exponent satisfies $\delta_{\rho}(\hat{\omega}_{\alpha})\le 1$, with equality only when the semisimple part of the Zariski closure is conjugate to the diagonal embedding $\rho_{D}(SL(2,\mathbb{R}))$, yielding a strong entropy rigidity statement both in the Poincaré series setting and the associated symmetric space setting via $\delta_{X}(\rho)$. The Manhattan-curve argument and a $C^1$ rigidity result for maximal representations are derived as corollaries, using the developed thermodynamic formalism and the hypertransversal structure. Together these results provide a robust rigidity picture for maximal representations in high rank, connecting growth rates, limit maps, and geometric structures through Patterson–Sullivan theory and Benoist’s cone framework. The methods have potential implications for higher Teichmüller theory and geometric group theory, offering precise criteria for when maximal representations coincide with the classical Fuchsian locus.

Abstract

Let $Γ\subset \mathsf{PSL}(2,\mathbb{R})$ be a lattice and $ρ:Γ\to \mathsf{Sp}(2n,\mathbb{R})$ be a maximal representation. We show that $ρ$ satisfies a measurable $(1,1,2)-$hypertransversality condition. With this we define a measurable Gromov product and the Bowen-Margulis-Sullivan measure associated to the unstable Jacobian introduced by Pozzetti, Sambarino and Wienhard. As a main application, we prove a strong entropy rigidity result for $ρ$.

Theorems & Definitions (65)

• Theorem 1.1
• Corollary 1.2
• proof
• Corollary 1.3
• proof
• Theorem 1.4
• Lemma 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4