Entropy Rigidity for Maximal Representations

Abstract

Let $\Gamma \subset \mathsf{PSL}(2,\mathbb{R})$ be a lattice and $\rho :\Gamma \to \mathsf{Sp}(2n,\mathbb{R})$ be a maximal representation. We show that $\rho$ 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 $\rho$.

arXiv commentRemoved an incorrect secondary result from the previous version. Corrected an imprecision in the statement of the main theorem. Made several other minor corrections