Entropy Rigidity for Maximal Representations

TL;DR

This paper establishes a rigidity property for maximal representations of lattices into symplectic groups, linking hypertransversality, Gromov products, and entropy to reveal structural constraints.

Contribution

It introduces a measurable hypertransversality condition and applies it to prove a strong entropy rigidity result for maximal representations.

Findings

Maximal representations satisfy a measurable hypertransversality condition.

A measurable Gromov product and Bowen-Margulis-Sullivan measure are constructed.

Entropy rigidity for maximal representations is established.

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 $\rho$.